GIFT  OF 
BOHEMIAN  CLUB 


From  the  collection  of  the 


V 


*        m 

Prelinger 

•      a 


library 


t 


San  Francisco,  California 
2006 


AMERICAN   SCIENCE  SEMES, 

FOB   HIGH   SCHOOLS    AND  COLLEGES. 


The  principal  objects  are  to  supply  the  lack — in  some  sub- 
jects very  great — of  authoritative  books  whose  principles  are, 
so  far  as  practicable,  illustrated  by  familiar  American  facts, 
and  also  to  supply  the  other  lack  that  the  advance  of  Science 
perennially  creates,  of  text-books  which  at  least  do  not  contra- 
dict the  latest  generalizations. 

In  large  12mo  volumes  of  about  500  pages  each. 

The  books  thus  far  published  or  arranged  for  are  as  follows  : 


/.  Astronomy. 

By  SIMON  NEWCOMB,  Supt. 
American  Nautical  Almanac, 
and  EDWARD  S.  HOLDEN, 
Professor  in  the  United  States 
Naval  Observatory.  12mo, 
$3.50. 

II.  Zoology. 

By  A.  S.  PACKARD,  Jr., 
Professor  of  Zoology  and  Ge- 
ology in  Brown  University, 
Editor  of  the  American  Nat- 
uralist. (In  November.) 

III.  Botany. 

By  C.  E.  BESSEY,  Professor 
in  the  Iowa  Agricultural  Col- 
lege and  late  Lecturer  in  the 
University  of  California.  (In 

Press.} 

IV.  Chemistry. 

By  SAMUEL  W.  JOHNSON 
and  WILLIAM  G.  MIXTER, 
Professors  in  Yale  College. 


V.  Physics. 

By  ALFRED  M.  MAYER,  Pro- 
fessor in  the  Stevens  Institute 
of  Technology,  nnd  ARTHUR 
W.  WRIGHT,  Professor  in 
Yale  College. 

VI.  Geology. 

By  RAPHAEL  PUMFELLY, 
late  Professor  in  Harvard 
University. 

VII.  The  Human  Body. 

By  H.  NEWELL  MARTIN, 
Professor  in  the  Johns  Hop- 
kins University. 

VIII.  Psychology. 

By  WILLIAM  JAMES,  Pro- 
fessor in  Harvard  University. 

IX.  Political  Economy. 

By  FRANCIS  A.  WALKER, 
Professor  in  Yale  College. 

X.  Government. 

By  EDWIN  L.  GODKIN,  Ed- 
itor of  the  Nation. 


HENRY  HOLT   &   CO.,  PUBLISHERS,  NEW  YORK. 


THE  PLANET   JUPITER. 
AB  seen  with  the  26-inch  telescope  at  Washington,  1875,  June  24. 


AMERICAN  SCIENCE  SERIES 


A8TEONOMT 


FOR 


SCHOOLS 


BY 

SIMON  NEWCOMB,    LL.D., 

SUPERINTENDENT    AMERICAN    EPHEMERIS    AND    NAUTICAL   ALMANAC, 
AND 

EDWARD  S.   HOLDEN,   M.A., 

PROFESSOR    IN    THE    U.   S.    NAVAL    OBSERVATORY. 


NEW    YORK 
HENRY  HOLT  AND   COMPANY 

1879 


OB  43 
K/67 


f 


Copyright,  1879, 

BY 

HENRY  HOLT  &  Co. 


PRESS  OF  JOHN  A.  GRAY,  AOT., 
18  JACOB  STREET, 

NEW  YORK. 


PREFACE. 


THE  following  work  is  designed  principally  for  the  use 
of  those  who  desire  to  pursue  the  study  of  Astronomy  as  a 
branch  of  liberal  education.  To  facilitate  its  use  by  stu- 
dents of  different  grades,  the  subject-matter  is  divided  into 
two  classes,  distinguished  by  the  size  of  the  type,  and  the 
volume  is  thus  made  to  contain  two  courses. 

The  portions  in  large  type  form  a  complete  course  for 
the  use  of  those  who  desire  only  such  a  general  knowledge 
of  the  subject  as  can  be  acquired  without  the  application 
of  advanced  mathematics.  It  is  believed  that  this  course 
can  be  mastered  by  persons  having  at  command  only  those 
geometrical  ideas  which  are  familiar  to  most  intelligent 
students  in  our  advanced  schools ;  though  sometimes,  es- 
pecially in  the  earlier  chapters,  a  knowledge  of  elementary 
trigonometry  and  physics  will  be  found  conducive  to  a 
full  understanding  of  a  few  details. 

The  portions  in  small  type  comprise  additions  for  the 
use  of  those  students  who  either  desire  a  more  detailed 
and  precise  knowledge  of  the  subject,  or  who  intend  to 
make  astronomy  a  special  study.  In  this,  as  in  the  ele- 
mentary course,  the  rule  has  been  never  to  use  more  ad- 
vanced mathematical  methods  than  are  necessary  to  the 
development  of  the  subject,  but  in  some  cases  a  knowl- 
edge of  Analytic  Geometry,  in  others  of  the  Differential 
Calculus,  and  in  others  of  Elementary  Mechanics,  isneces- 

701050 


x  CONTENTS. 

CHAPTER  IX. 

PAGE 

THE  PLANET  URANUS — Satellites  of  Uranus 362 

CHAPTER  X. 
THE  PLANET  NEPTUNE — Satellite  of  Neptune 365 

CHAPTER  XI. 
THE  PHYSICAL  CONSTITUTION  OP  THE  PLANETS 370 

CHAPTER  XII. 

METEORS. 

Phenomena  and  Causes  of  Meteors — Meteoric  Showers. .    375 

CHAPTER   XIII. 

COMETS. 

Aspect  of  Comets — The  Vaporous  Envelopes — The  Physical  Con- 
stitution of  Comets — Motion  of  Comets — Origin  of  Comets — 
Remarkable  Comets. .  .  388 


PART    III. 

THE   UNIVERSE   AT   LARGE. 


INTRODUCTION 411 

CHAPTER  I. 

THE  CONSTELLATIONS. 

General  Aspect  of  the  Heavens— Magnitude  of  the  Stars — The 
Constellations  and  Names  of  the  Stars — Description  of  Con- 
stellations— Numbering  and  Cataloguing  the  Stars 415 

CHAPTER  II. 

VARIABLE  AND  TEMPORARY  STARS. 

Stars  Regularly  Variable— Temporary  or  New  Stars— Theory  of 
Variable  Stars..  440 


CONTENTS.  xi 

CHAPTER  III. 

MULTIPLE    STABS. 

PAGK 

Character  of  Double  and  Multiple  Stars — Orbits  of  Binary  Stars. .  448 
CHAPTER   IV. 

NEBULAE   AND   CLUSTERS. 

Discovery  of  Nebulae — Classification  of  Nebulae  and  Clusters — 
Star  Clusters — Spectra  of  Nebulae  and  Clusters — Distribution 
of  Nebulae  and  Clusters  on  the  Surface  of  the  Celestial 
Sphere 457 

CHAPTER  V. 

SPECTRA   OF   FIXED   STARS. 

Characters  of  Stellar  Spectra — Motion  of  Stars  in  the  Line  of  Sight.  468 
CHAPTER  VI. 

MOTIONS   AND   DISTANCES   OF   THE    STARS. 

Proper  Motions — Proper  Motion   of    the    Sun — Distances   of  the 

Fixed  Stars 472 

CHAPTER  VII. 
CONSTRUCTION  OF  THE  HEAVENS 478 

CHAPTER  VIII. 
COSMOGONY 492 

Index..  .  503 


ASTRONOMY 


INTRODUCTION. 

ASTRONOMY  (otGrrjp — a  star,  and  VO'/AOS — a  law)  is  the 
science  which  has  to  do  with  the  heavenly  bodies,  their 
appearances,  their  nature,  and  the  laws  governing  their 
real  and  their  apparent  motions. 

In  approaching  the  study  of  this,  the  most  ancient  of  the 
sciences  depending  upon  observation,  it  must  be  borne  in 
mind  that  its  progress  is  most  intimately  connected  with 
that  of  the  race,  it  having  always  been  the  basis  of  geog- 
raphy and  navigation,  and  the  soul  of  chronology.  Some 
of  the  chief  advances  and  discoveries  in  abstract  mathe- 
matics have  been  made  in  its  service,  and  the  methods 
both  of  observation  and  analysis  once  peculiar  to  its  prac- 
tice now  furnish  the  firm  bases  upon  which  rest  that  great 
group  of  exact  sciences  which  we  call  physics. 

It  is  more  important  to  the  student  that  he  should  be- 
come penetrated  with  the  spirit  of  the  methods  of  astron- 
omy than  that  he  should  recollect  its  minutiae,  and  it  is 
most  important  that  the  knowledge  which  he  may  gain 
from  this  or  other  books  should  be  referred  by  him  to  its 
true  sources.  For  example,  it  will  often  be  necessary  to 
speak  of  certain  planes  or  circles,  the  ecliptic,  the  equa- 
tor, the  meridian,  etc. ,  and  of  the  relation  of  the  appa- 
rent positions  of  stars  and  planets  to  them  ;  but  his  labor 
will  be  useless  if  it  has  not  succeeded  in  giving  him  a 
precise  notion  of  these  circles  and  planes  as  they  exist  in 


ASTRONOMY. 


the  sky,  and  not  merely  in  the  figures  of  his  text-book. 
Above  all,  the  study  of  this  science,  in  which  not  a  single 
step  could  have  been  taken  without  careful  and  painstak- 
ing observation  of  the  heavens,  should  lead  its  student 
himself  to  attentively  regard  the  phenomena  daily  and 
hourly  presented  to  him  by  the  heavens. 

Does  the  sun  set  daily  in  the  same  point  of  the  hori- 
zon ?  Does  a  change  of  his  own  station  affect  this  and 
other  aspects  of  the  sky  ?  At  what  time  does  the  full 
moon  rise  ?  Which  way  are  the  horns  of  the  young 
moon  pointed  ?  These  and  a  thousand  other  questions 
are  already  answered  by  the  observant  eyes  of  the  an- 
cients, who  discovered  not  only  the  existence,  but  the 
motions,  of  the  various  planets,  and  gave  special  names  to 
no  less  than  fourscore  stars.  The  modern  pupil  is  more 
richly  equipped  for  observation  than  the  ancient  philoso- 
pher. If  one  could  have  put  a  mere  opera-glass  in  the 
hands  of  HIPPAKCHUS  the  world  need  not  have  waited  two 
thousand  years  to  know  the  nature  of  that  early  mystery, 
the  Milky  Way,  nor  would  it  have  required  a  GALILEO  to 
discover  the  phases  of  Venus  and  the  spots  on  the  sun. 

From  the  earliest  times  the  science  has  steadily  progress- 
ed by  means  of  faithful  observation  and  sound  reasoning 
upon  the  data  which  observation  gives.  The  advances  in 
our  special  knowledge  of  this  science  have  made  it  con- 
venient to  regard  it  as  divided  into  certain  portions,  which 
it  is  often  convenient  to  consider  separately,  although  the 
boundaries  cannot  be  precisely  fixed. 

Spherical  and  Practical  Astronomy.  —  First  in  logical 
order  we  have  the  instruments  and  methods  by  which  the 
positions  of  the  heavenly  bodies  are  determined  f  roin  obser- 
vation, and  by  which  geographical  positions  are  also  fixed. 
The  branch  which  treats  of  these  is  called  spherical  and 
practical  astronomy.  Spherical  astronomy  provides  the 
mathematical  theory,  and  practical  astronomy  (which  is 
almost  as  much  an  art  as  a  science)  treats  of  the  applica- 
tion of  this  theory. 


DIVISIONS  OF  THE  SUBJECT.  3 

Theoretical  Astronomy  deals  with  the  laws  of  motion  of 
the  celestial  bodies  as  determined  by  repeated  observations 
of  their  positions,  and  by  the  laws  according  to  which  they 
ought  to  move  under  the  influence  of  their  mutual  gravi- 
tation. The  purely  mathematical  part  of  the  science,  by 
which  the  laws  of  the  celestial  motions  are  deduced  from 
the  theory  of  gravitation  alone,  is  also  called  Celestial 
Mechanics,  a  term  first  applied  by  LA  PLACE  in  the  title  of 
his  great  work  Mecanique  Celeste. 

Cosmical  Physics. — A  third  branch  which  has  received 
its  greatest  developments  in  quite  recent  times  may  be 
called  Cosmical  Physics.  Physical  astronomy  might  be 
a  better  appellation,  were  it  not  sometimes  applied  to 
celestial  mechanics.  This  branch  treats  of  the  physical 
constitution  and  aspects  of  the  heavenly  bodies  as  investi- 
gated with  the  telescope,  the  spectroscope,  etc. 

We  thus  have  three  great  branches  which  run  into  each 
other  by  insensible  gradations,  but  under  which  a  large 
part  of  the  astronomical  research  of  the  present  day  may 
be  included.  In  a  work  like  the  present,  however,  it 
will  not  be  advisable  to  follow  strictly  this  order  of  sub- 
jects ;  we  shall  rather  strive  to  present  the  whole  subject 
in  the  order  in  which  it  can  best  be  understood.  This 
order  will  be  somewhat  like  that  in  which  the  knowl- 
edge has  been  actually  acquired  by  the  astronomers  of 
different  ages. 

Owing  to  the  frequency  with  which  we  have  to  use 
terms  expressing  angular  measure,  or  referring  to  circles 
on  a  sphere,  it  may  be  admissible,  at  the  outset,  to  give 
an  idea  of  these  terms,  and  to  recapitulate  some  prop- 
erties of  the  sphere. 

Angular  Measures. — The  unit  of  angular  measure  most 
used  for  considerable  angles,  is  the  degree,  360  of  which 
extend  round  the  circle.  The  reader  knows  that  it  is  90° 
from  the  horizon  to  the  zenith,  and  that  two  objects  180° 
apart  are  diametrically  opposite.  An  idea  of  distances  of 


4  ASTRONOMY. 

a  few  degrees  may  be  obtained  by  looking  at  the  two  stars 
which  form  the  pointers  in  the  constellation  Ursa  Major 
(the  Dipper),  soon  to  be  described.  These  stars  are  5° 
apart.  The  angular  diameters  of  the  sun  and  moon  are 
each  a  little  more  than  half  a  degree,  or  30'. 

An  object  subtending  an  angle  of  only  one  minute  ap- 
pears as  a  point  rather  than  a  disk,  but  is  still  plainly  vis- 
ible to  the  ordinary  eye.  HELMHOLTZ  finds  that  if  two 
minute  points  are  nearer  together  than  about  V  12",  no 
eye  can  any  longer  distinguish  them  as  two.  If  the  ob- 
jects are  not  plainly  visible — if  they  are  small  stars,  for 
instance,  they  may  have  to  be  separated  3',  5',  or  even 
10',  to  be  seen  as  separate  objects.  Near  the  star  a  Lyrce 
are  a  pair  of  stars  3%'  apart,  which  can  be  separated  only 
by  very  good  eyes. 

If  the  object  be  not  a  point,  but  a  long  line,  it  may  be 
seen  by  a  good  eye  when  its  breadth  subtends  an  angle  of 
only  a  fraction  of  a  minute  ;  the  limit  probably  ranges 
from  10"  to  15". 

If  the  object  be  much  brighter  than  the  background  on 
which  it  is  seen,  there  is  no  limit  below  which  it  is  neces- 
sarily invisible.  Its  visibility  then  depends  solely  on  the 
quantity  of  light  which  it  sends  to  the  eye.  It  is  not 
likely  that  the  brightest  stars  subtend  an  angle  of  yj-^  of 
a  second. 

So  long  as  the  angle  subtended  by  an  object  is  small,  we 
may  regard  it  as  varying  directly  as  the  linear  magnitude 
of  the  body,  and  inversely  as  its  distance  from  the  ob- 
server. A  line  seen  perpendicularly  subtends  an  angle 
of  1°  when  it  is  a  little  less  than  60  times  its  length  dis- 
tant from  the  observer  (more  exactly  when  it  is  57-3 
lengths  distant) ;  an  angle  of  V  when  it  is  3438  lengths 
distant,  and  of  1"  when  it  is  206265  lengths  distant. 
These  numbers  are  obtained  by  dividing  the  number  of 
degrees,  minutes,  and  seconds,  respectively,  in  the  cir- 
cumference, by  2  x  3-14159265,  the  ratio  of  the  circum- 
ference of  a  circle  to  the  radius. 


CIRCLES  OF  THE  SPHERE.  5 

Great  Circles  of  the  Sphere. — In  Fig.  1  let  the  outline 
represent  that  of  a  sphere,  around  which  are  described 
the  two  great  circles  A  EB  F  and  C  E  D  F.  These  cir- 
cles are  the  lines  in  which  two  planes  passing  through  the 
centre  0  of  the  sphere  intersect  the  latter.  We  may  con- 
sider them  as  representing  the  planes. 

The  points  P  and  P ',  each  of  which  is  90°  distant 
from  every  point  of  the  circle  A  E  E  F^  are  called  the 


FlG.    1. — SECTIONS   OF   A   SPHERE   BY  PLANES. 

poles  of  that,  circle.  The  poles  are  the  points  in  which  a 
hne  passing  through  the  centre  O  perpendicular  to  the 
plane  of  the  circle  meets  the  sphere.  They  may  be  con- 
sidered as  representing  this  line. 

The  angle  B  D,  or  A  C,  equal  to  the  greatest  distance 
of  the  two  circles,  is  the  same  as  the  angle  which  the 
planes  of  the  circles  make  with  each  other.  The  dis- 
tance between  the  poles  P  Q  or  P'  Q'  is  equal  to  the  same 
angle.  There  are  therefore  three  equivalent  representa- 
tives for  what  may  be  considered  the  same  element ; 
namely  :  (1)  the  inclination  of  the  planes  of  two  circles  ; 
(2)  the  angle  between  their  poles  ;  and  (3)  the  greatest 
angles,  A  C  or  B  D,  between  the  circles  on  the  celestial 
sphere. 


SYMBOLS  AND  ABBREVIATIONS. 


SIGNS  OF  THE  PLANETS,   ETC. 


0 

• 

$ 

0  or  $ 


The  Sun. 
The  Moon. 
Mercury. 
Venus. 
The  Earth. 


$  Mars. 

It  Jupiter. 

^  Saturn. 

(5  Uranus. 

tJJ  Neptune. 


The  asteroids  are  distinguished  by  a  circle  inclosing  a  number,  which 
number  indicates  the  order  of  discovery,  or  by  their  names,  or  by  both, 
as  100 


SIGNS  OP  THE  ZODIAC. 


Spring 

4ns? 

Summer 
81gDS- 


f  £ries' 
»   Taurus. 
n  Gemini. 

®  TCancer* 
S)   Leo. 

«  Virgo. 


Autumn 
signs. 


7. 
8. 
9. 


Libra. 
H,  Scorpius. 
#    Sagittarius. 


Winter  {  J°-  *  Capricornus. 

<  11.  £?  Aquarius. 
81gDS-    1 12.  K  Pisces. 


ASPECTS. 


(5   Conjunction,  or  having  the  same  longitude  or  right  ascension. 

n   Quadrature,  or  differing  90°  in 

8   Opposition,  or  differing  180°  in  "  " 


ASTRONOMICAL  SYMBOLS. 

MISCELLANEOUS   SYMBOLS. 


Q   Ascending  node. 
£3   Descending  node. 
N.  North.     S.  South. 
E.  East.     W.  West. 

0  Degrees. 

'  Minutes  of  arc. 

"  Seconds  of  arc. 

h  Hours. 

111  Minutes  of  time. 

"  Seconds  of  time. 
L,  Mean  longitude  of  a  body. 
g,  Mean  anomaly. 
f,  True  anomaly. 
n,  Mean  sidereal  motion  in  a  unit 

of  time. 

r,  Radius  vector. 
<j>,  Angle  of  eccentricity. 

TT,  Longitude   of    perihelion    (also  j  a,  

parallax).  !  A,  Azimuth. 

p,  Earth's  Equatorial  radius. 


JR. A,  or  a,  Right  ascension. 
Dec.  or  6,  Declination. 
C,  True  zenith  distance. 
£',  Apparent  zenith  distance. 
A  Distance  from  the  earth. 
I,  Heliocentric  longitude. 
b,  Heliocentric  latitude. 
/I,  Geocentric  longitude. 
/?,  Geocentric  latitude. 
6  or   i2,   Longitude  of    ascending 

node. 

i,  Inclination  of  orbit  to  the  eclip- 
tic. 

w,  Angular  distance  from   perihe- 
lion to  node. 

u,  Distance   from   node,   or   argu- 
ment for  latitude. 
Altitude. 


The  Greek  alphabet  is  here  inserted  to  aid  those  who  are  not  already 
familiar  with  it  in  reading  the  parts  of  the  text  in  which  its  letters 
occur  : 


Letters. 

Names. 

A   a 

Alpha 

B  /?  6 

Beta 

r  rr 

Gamma 

A  d 

Delta 

E  e 

Epsilon 

z  a 

Z6ta 

H  ^ 

Eta 

e$Q 

Theta 

I  i 

Iota 

K  K 

Kappa 

AX 

Lambda 

M/i 

Mu 

Letters. 

Names. 

N  v 

Nu 

H  £ 

Xi 

O  o 

Omicron 

11  n  TT 

Pi 

P  pq 

Rho 

2    a  5 

Sigma 

T  rl 

Tau 

TV 

Upsilon 

4>  0 

Phi 

x* 

Chi 

f* 

Psi 

flw 

Omega 

THE  METRIC  SYSTEM. 

THE  metric  system  of  weights  and  measures  being  employed  in 
this  volume,  the  following  relations  between  the  units  of  this  system 
most  used  and  those  of  our  ordinary  one  will  be  found  convenient  for 
reference  : 

MEASURES  OF  LENGTH. 

1  kilometre    =  1000  metres        =    0-62137  mile. 
1  metre          =  the  unit  =  39-37  inches. 

1  millimetre  =  y^  of  a  metre  =    0-03937  inch. 


MEASURES  OF  WEIGHT. 


1  millier  or  tonneau  =  1,000,000  grammes  =  2204-6  pounds. 
1  kilogramme  1000  grammes  =        2-2046  pounds. 

1  gramme  =  the  unit  =      15-432  grains. 

1  milligramme  =  y^j  &  of  a  gramme    =        0-01543  grain. 


The  following  rough  approximations  may  be  memorized  : 

The  kilometre  is  a  little  more  than  1%  of  a  mile,  but  less  than  £  of 
a  mile. 

The  mile  is  1-ffr  kilometres. 

The  kilogramme  is  2£  pounds. 

The  pound  is  less  than  half  a  kilogramme. 


CHAPTER  I. 

THE  RELATION  OF  THE  EARTH  TO  THE 
HEAVENS. 

§  1.  THE  EARTH. 

IN  considering  the  relation  of  the  earth  to  the  heavens, 
we  necessarily  begin  with  the  earth  itself  ;  not  simply 
because  we  now  know  it  to  be  one  of  the  heavenly  bodies, 
but  because  it  is  from  its  surface  that  all  observations  of 
the  heavens  have  to  be  made. 

A  consideration  of  well-known  facts  will  show  that  this 
earth  upon  which  we  live  is,  at  least  approximately,  a 
globe  whose  dimensions  are  gigantic 
when  compared  to  our  ordinary  and 
daily  ideas  of  size.  Its  shape  is  in 
several  ways  known  to  be  nearly 
that  of  a  sphere. 

I.  It  has  been  repeatedly  circum- 
navigated in  various  directions. 

II.  Portions  of  its  surface,  visi- 
ble from  elevated  positions  in  the 
midst  of  extensive  plains  or  at  sea, 

appear  to  be  bounded  by  circles.  FIG.  2. 

This  appearance  at  all  points  of  the     illustrating  the  fact  that  the 

.,          r   f         ,       ,      .  .      .,    portions  of   the    earth  visible 

Sill-face    Of    a    body  IS    a    geometrical   from  elevated  positions,  8,  S>, 

..,  f          ,    , i     ,        f  ,  S",  etc. ,  are  bounded  by  circles. 

attribute  of  a  globular  form  only. 

III.  Further  than  this  we  know  that  careful  measure- 
ments of  portions  of  the  globe  by  the  various  national 
geodetic  surveys  have  agreed  with  this  general  conclusion. 


10  ASTRONOMY. 

More  precise  reasons  will  be  apparent  later,  but  these  will 
be  sufficient  to  base  our  general  considerations  upon.  Of 
the  size  of  the  earth  we  may  form  a  rough  idea  by  the 
time  required  to  travel  completely  around  it,  which  is 
now  about  three  months. 

We  find  next  that  this  globe  is  completely  isolated 
in  space.  It  neither  rests  on  any  thing  else,  nor  is  it  in 
contact  with  any  surrounding  body.  The  most  obvious 
proof  of  this  which  presents  itself  is,  that  mankind  have 
visited  nearly  every  part  of  its  surface  without  finding 
any  such  connection,  and  that  the  heavenly  bodies  seem 
to  perform  complete  circuits  around  it  and  under  it  with- 
out meeting  with  any  obstacles.  The  sun  which  rose  to- 
day is  the  same  body  as  the  setting  sun  of  yesterday,  but 
it  has  been  seen  to  move  (apparently)  about  the  earth 
from  east  to  west  during  the  day,  and  it  regularly  reap- 
pears each  morning.  Moreover,  if  attentively  watched, 
it  will  be  found  to  rise  and  set  at  different  parts  of  the 
horizon  of  any  place  at  different  times  of  the  year,  which 
negatives  the  ancient  belief  that  its  nocturnal  journey  was 
made  through  a  huge  subterranean  tunnel. 

§  2.  THE  DIURNAL  MOTION  AND  THE  CELESTIAL 
SPHERE. 

Passing  now  from  the  earth  to  the  heavens,  and  viewing 
the  sun  by  day,  or  the  stars  by  night,  the  first  phenomenon 
which  claims  our  attention  is  that  of  the  diurnal  motion. 

"We  must  here  caution  the  reader  to  carefully  distin- 
guish between  apparent  and  real  motions.  For  example, 
when  the  phenomena  of  the  diurnal  motion  are  set  forth 
as  real  visible  motions,  he  must  be  prepared  to  learn  sub- 
sequently that  this  appearance,  which  is  obvious  to  all,  is 
yet  a  consequence  of  a  real  motion  only  to  be  detected  by 
reason.  "We  shall  first  describe  the  diurnal  motion  as  it 
appears,  and  show  that  all  the  appearances  to  a  spectator 
at  any  one  place  may  be  represented  by  supposing  the 
earth  to  remain  fixed  in  space,  and  the  whole  concave  of 


THE  DIURNAL  MOTION.  11 

the  neavens  to  turn  about  it,  and  finally  it  will  be  shown 
that  we  have  reason  to  believe  that  the  solid  earth  itself 
is  in  constant  rotation  while  the  heavens  remain  immov- 
able, presenting  different  portions  in  turn  to  the  observer. 
The  motion  in  question  is  most  obvious  in  the  case  of  the 
sun,  which  appears  to  make  a  daily  circuit  in  the  heavens, 
rising  in  the  east,  passing  over  toward  the  south,  setting  in 
the  west,  and  moving  around  under  the  earth  until  it 
reaches  the  eastern  horizon  again.  Observations  of  the  stars 
made  through  any  one  evening  show  that  they  also  appear 
to  perform  a  similar  circuit.  Whatever  stars  we  see  near 
the  eastern  horizon  will  be  found  constantly  rising  higher, 
and  moving  toward  the  south,  while  those  in  the  west 
will  be  constantly  setting.  If  we  watch  a  star  which  is 
rising  at  the  same  point  of  the  horizon  where  the  sun 
rises,  we  shall  find  it  to  pursue  nearly  the  same  course  in 
the  heavens  through  the  night  that  the  sun  follows 
through  the  day.  Continued  observations  will  show, 
however,  that  there  are  some  stars  which  do  not  set  at  all — 
namely,  those  in  the  north.  Instead  of  rising  and  setting, 
they  appear  to  perform  a  daily  revolution  around  a  point 
in  the  heavens  which  in  our  latitudes  is  nearly  half  way 
between  the  zenith  and  the  northern  horizon.  This  cen- 
tral point  is  called  the  pole  of  the  heavens.  Near  it  is 
situated  Polaris,  or  the  pole  star.  It  may  be  recog- 
nized by  the  Pointers,  two  stars  in  the  constellation 
Ursa  Major,  familiarly  known  as  The  Dipper.  These 
stars  are  shown  in  Fig.  3.  If  we  watch  any  star  be- 
tween the  pole  and  the  north  horizon,  we  shall  find 
that  instead  of  moving  from  east  to  west,  as  the  stars 
generally  appear  to  move,  it  really  appears  to  move 
toward  the  east  ;  but  instead  of  continuing  its  motion  and 
setting  in  the  east,  we  shall  find  that  it  gradually  curves 
its  course  upward.  If  we  could  follow  it  for  twenty-four 
hours  we  should  see  it  move  upwards  in  the  north-east,  and 
then  pass  over  toward  the  west  between  the  zenith  and 
the  pole,  then  sink  down  in  the  north-west ;  and  on  the 


12  ASTRONOMY. 

following  night  curve  its  course  once  more  toward  the 
east.  The  arc  which  it  appears  to  describe  is  a  perfect 
circle,  having  the  pole  in  its  centre.  The  farther  from 
the  pole  we  go,  the  larger  the  circle  which  each  star  seems 
to  describe  ;  and  when  we  get  to  a  distance  equal  to  that 
between  the  pole  and  the  horizon,  each  star  in  its  appa- 
rent passage  below  the  pole  just  grazes  the  horizon. 


FlG.    3. — THE  APPARENT   DIURNAL   MOTION. 

As  a  result  of  this  apparent  motion,  each  individual 
constellation  changes  its  configuration  with  respect  to  the 
horizon,  that  part  which  is  highest  when  the  constellation 
is  above  the  pole  being  lowest  when  below  it.  This  is 
shown  in  Figure  4,  which  represents  a  supposed  constel- 
lation at  five  different  times  of  the  night. 

Going  farther  still  from  the  pole,  the  stars  will  dip  be- 


THE  DIURNAL  MOTION. 


13 


low  the  horizon  during  a  portion  of  their  course,  and  the 
fraction  of  the  circle  which  is  below  the  pole  will  be  con- 
tinually increasing.  Looking  yet  farther  south  we  shall 
find  one  half  of  the  circle  to  be  above  and  one  half  below 
the  horizon.  Farther  yet,  we  shall  find  the  stars  describing 
shorter  arcs  while  above  the  horizon,  and  therefore  longer 
ones  below  it.  Near  the  south  horizon,  each  star  rises 
for  only  a  short  time  a  little  to  the  east  of  south,  and  soon 
sets  a  little  to  the  west  of  it. 


NORTH 

FIG.  4. 

If  we  carefully  study  this  motion,  we  shall  find  that  it 
does  not  arise  from  each  star  pursuing  an  independent 
course,  for  not  only  do  all  the  stars  perform  this  ap- 
parent revolution  in  the  very  same  time,  but  they  also 
preserve  unchanged  their  relative  distances  from  each 
other,  with  tjie  exception  of  five,  called  planets  or  wan- 
dering stars.  The  thousands  of  others  which  are  visible 
to  the  naked  eye  preserve  their  relative  positions  with 
such  exactness  that  the  ordinary  observer  could  perceive 
no  change  even  after  the  lapse  of  centuries.  This  fact 
naturally  suggested  to  the  ancients  the  idea  that  there 
must  be  some  material  connection  between  the  stars.  An 


14  ASTRONOMY. 

apparent  explanation,  both  o£  this  and  of  the  phenomena 
of  the  diurnal  motion,  was  offered  by  the  conception  of 
the  celestial  sphere.  The  salient  phenomena  of  the 
heavens,  from  whatever  point  of  the  earth's  surface  they 
might  be  viewed,  were  represented  by  supposing  that  the 
globe  of  the  earth  was  situated  centrally  within  an  im- 
mensely larger  hollow  sphere  of  the  heavens.  The  vis- 
ible portion,  or  upper  half  of  this  hollow  sphere,  as  seen 
from  any  point,  constituted  the  celestial  vault,  and  the 
whole  sphere,  with  the  stars  which  studded  it,  was  called 
the  firmament.  The  stars  were  set  in  its  interior  surface, 
or  the  firmament  might  be  supposed  to  be  of  a  perfectly 
transparent  crystal,  and  the  stars  might  be  situated  in  any 
portion  of  its  thickness.  About  one  half  of  the  sphere 
could  be  seen  from  any  point  of  the  earth's  surface,  the 
view  of  the  other  half  being  necessarily  cut  off  by  the 
earth  itself.  This  sphere  was  conceived  to  make  a  diurnal 
revolution  around  an  axis,  necessarily  a  purely  mathemat- 
ical line,  passing  centrally  through  it  and  through  the 
earth.  The  ends  of  this  axis  were  the  poles.  The  situa- 
tion of  the  north  end,  or  north  pole,  was  visible  in  north- 
ern latitudes,  while  the  south  pole  was  invisible,  being 
below  the  horizon.  A  navigator  sailing  south  would  so 
change  his  horizon,  owing  to  the  sphericity  of  the  earth, 
that  the  location  of  the  north  pole  would  sink  out  of  sight, 
while  that  of  the  south  pole  would  come  into  view. 

It  was  clearly  seen,  even  by  the  ancients,  that  the  diur- 
nal motion  could  be  as  well  represented  by  supposing  the 
celestial  sphere  to  be  at  rest,  and  the  earth  to  revolve 
around  this  axis,  as  by  supposing  the  sphere  to  revolve. 
This  doctrine  of  the  earth's  rotation  was  maintained  by 
several  of  the  ancient  astronomers,  notably  by  ARISTAR- 
CHUS  and  TIMOCHARIS.  The  opposite  view,  however,  was 
maintained  by  PTOLEMY,  who  could  not  conceive  that  the 
earth  could  be  endowed  with  such  a  rapid  rotation  with- 
out disturbing  the  motion  of  bodies  at  its  surface.  We 
now  know  that  PTOLEMY  was  wrong,  and  his  opponents 


THE  CELESTIAL  SPHERE.  15 

right.  Still,  so  far  as  the  apparent  diurnal  motion  is  con- 
cerned, it  is  indifferent  whether  we  conceive  the  earth  or 
the  heavens  to  be  in  motion.  Sometimes  the  one  concep- 
tion, and  sometimes  the  other,  will  make  the  phenomena 
the  more  clear.  As  a  matter  of  fact,  astronomers  speak 
of  the  sun  rising  and  setting,  just  as  others  do,  although 
it  is  in  reality  the  earth  which  turns.  This  is  a  form  of 
language  which,  being  designed  only  to  represent  the  ap- 
pearances, need  not  lead  us  into  error. 

The  celestial  sphere  which  we  have  described  has  long 
ceased  to  iigure  in  astronomy  as  a  reality.  We  now  know 
that  the  celestial  spaces  are  practically  perfectly  void  ; 
that  some  of  the  heavenly  bodies,  which  appear  to  be  on 
the  surface  of  the  celestial  sphere  at  equal  distances  from 
the  earth  as  a  centre,  are  thousands,  or  even  millions  of 
times  farther  from  the  earth  than  others  ;  that  there  is  no 
material  connection  between  them,  and  that  the  celestial 
sphere  itself  is  only  a  result  of  optical  perspective.  But 
the  language  and  the  conception  are  still  retained,  because 
they  afford  the  most  clear  and  definite  method  of  repre- 
senting the  directions  of  the  heavenly  bodies  from  the 
observer,  wherever  he  may  be  situated.  In  this  respect 
it  serves  the  same  purpose  that  the  geometric  sphere 
does  in  spherical  trigonometry.  The  student  of  this  sci- 
ence knows  that  there  is  really  no  need  of  supposing  a 
sphere  or  a  spherical  triangle,  because  every  spherical  arc 
is  only  the  representative  of  an  angle  between  two  lines 
which  emanate  from  the  centre,  one  to  each  end  of  the 
arc,  while  the  angles  of  the  triangle  are  only  those  of  the 
planes  containing  the  three  lines  which  are  drawn  to 
each  angle  from  the  centre.  Spherical  trigonometry  is, 
therefore,  in  reality,  only  the  trigonometry  01  solid 
angles  ;  and  the  purpose  of  the  sphere  is  only  to  afford  a 
convenient  method  of  conceiving  of  such  angles.  In  the 
same  way,  although  the  celestial  sphere  has  no  real  ex- 
istence, yet  by  conceiving  of  it  as  a  reality,  and  supposing 
certain  lines  of  reference  drawn  upon  it,  we  are  enabled  to 


16 


ASTRONOMY. 


form  an  idea  of  the  relative  directions  of  the  heavenly 
bodies.  We  may  conceive  of  it  in  two  ways  :  firstly,  as 
having  an  infinite  radius,  in  which  case  the  centre  of  the 
earth,  or  any  point  of  its  surface,  may  equally  be  supposed 
to  be  in  the  centre  of  the  celestial  sphere  ;  or,  secondly,  we 
may  suppose  it  to  be  finite,  the  observer  carrying  the  cen- 


FlG.    5. — STABS   SEEN   ON    THE   CELESTIAL    SPHERE. 

tre  with  him  wherever  he  goes.  The  first  assumption  will 
probably  be  the  one  which  it  is  best  to  adopt.  The  object 
attained  by  each  mode  of  representation  is  that  of  having 
the  observer  always  in  the  centre  of  the  supposed  sphere. 
Fig.  5  will  give  the  reader  an  idea  of  its  application.  He 
is  supposed  to  be  stationed  in  the  centre,  0,  and  to  have 
around  him  the  bodies  p  qr  s  t,  etc.  The  sphere  itself 
being  supposed  at  an  immense  distance,  outside  of  all 
these  bodies,  we  may  suppose  lines  to  be  drawn  from 
each  of  them  directly  away  from  the  centre  until  they 
reach  the  sphere.  The  points  P  Q  R  S  T,  etc. ,  in  which 


THE  CELESTIAL  SPHERE.  17 

these  lines  intersect  the  sphere,  will  represent  the  appa- 
rent positions  of  the  heavenly  bodies  as  seen  by  the  ob- 
server at  0.  If  several  of  them,  as  those  marked  tt  t, 
are  in  the  same  direction  from  the  observer,  they  will  ap- 
pear to  be  projected  on  the  same  point  of  the  sphere. 
Thus  positions  on  the  sphere  represent  simply  the  direc- 
tions in  which  the  bodies  are  seen,  but  have  no  direct  re- 
lations to  the  distance. 

It  was  seen  by  the  ancients  that  the  earth  was  only  a 
point  in  comparison  with  the  apparent  sphere  of  the  fixed 
stars.  This  was  shown  by  the  uniformity  of  the  diurnal 
motion  ;  if  the  earth  had  any  sensible  magnitude  in  com- 
parison with  the  sphere  of  the  heavens,  the  sun,  or  a  star, 
would  seem  to  be  nearer  to  the  observer  when  it  passed 
the  meridian,  or  any  point  near  his  zenith,  than  it  would 
when  it  was  below  the  horizon,  or  nearly  under  his  feet, 
by  a  quantity  equal  to  the  diameter  of  the  earth.  Being 
nearer  to  him,  it  would  seem  to  move  more  rapidly  when 
above  the  horizon  than  when  below,  and  its  apparent  angular 
dimensions  would  be  greater  in  the  zenith  than  in  the 
horizon.  As  a  matter  of  fact,  however,  the  most  refined 
observations  do  not  show  the  slightest  variation  from 
perfect  uniformity,  no  matter  what  the  point  at  which 
the  observer  may  stand.  Therefore,  observers  all  over 
the  earth  are  apparently  equally  near  the  stars  at  every 
point  of  their  apparent  diurnal  paths ;  whence  their 
distance  must  be  so  great  that  in  proportion  to  them  the 
diameter  of  the  earth  entirely  vanishes.  This  argument 
holds  equally  true  whether  we  suppose  the  earth  or  the 
heavens  to  revolve,  because  the  observer,  carried  around 
by  the  rotating  earth,  will  be  brought  nearer  to  those 
stars  which  are  over  his  head,  and  carried  farther  from 
them  when  he  is  on  the  opposite  side  of  the  circle  in 
which  he  moves. 

Suppose  the  earth  to  be  at  0,  and  the  celestial  sphere  of  the  fixed 
stars  to  be  represented  in  the  figure  by  the  circle  N  Z  Q  S  n,  etc. 
Suppose  .ZV  E  S  W  to  represent  the  plane  of  the  horizon  of  some 


18 


ASTRONOMY. 


observer  on  the  earth's  surface,     tie  will  then  see  every  thing  above 

this  plane,  and  nothing  below  it. 
If  N  E  8  is  his  eastern  horizon, 
stars  will  appear  to  rise  at  various 
points,  gr,  E,  d,  a,  etc.,  and  will 
appear  to  describe  circles  until 
they  attain  their  highest  points 
at  ^,  Q,  e,  Z>,  etc.,  sinking  into 
the  western  horizon  at  &,  TT,  /,  c, 
etc.  These  are  facts  of  observa- 
tion. The  common  axis  of  these 
circles  is  P  p,  and  stars  about  P 
(the  pole}  never  set.  The  appa- 
rent diurnal  arc  I  m,  for  instance, 
represents  the  apparent  orbit  of 
a  circumpolar  star. 

FIG.  6. 


§  3.  CORRESPONDENCE    OP    THE    TERRESTRIAL 
AND    CELESTIAL    SPHERES. 

We  have  said  that  the  direction  of  a  heavenly  body 
from  an  observer,  or,  which  is  the  same  thing,  its  ap- 
parent position,  is  defined  by  the  point  of  the  celestial 
sphere  on  which  it  seems  to  be.  This  point  is  that  in 
which  the  straight  line  drawn  from  the  observer  to  the 
body,  and  continued  forward  indefinitely,  meets  the  celes- 
tial sphere.  Its  position  is  fixed  by  reference  to  certain 
fundamental  circles  supposed  to  be  drawn  on  the  sphere, 
on  the  same  plan  by  which  longitude  and  latitude  on  the 
earth  are  fixed.  The  system  of  thus  defining  terrestrial 
positions  by  reference  to  the  earth's  equator,  and  to  some 
prime  meridian  from  which  we  reckon  the  longitudes,  is  one 
with  which  the  reader  may  be  supposed  familiar.  We  shall 
therefore  commence  with  those  circles  of  the  celestial 
sphere  which  correspond  to  the  meridians,  parallels,  etc. , 
on  the  earth. 

First,  we  remark  that  if  we  consider  the  earth  to  be  at 
rest  for  a  moment,  every  point  on  its  surface  is  at  the  end 
of  a  radius  which,  if  extended,  would  touch  a  correspond- 


THE  CELESTIAL  AND   TERRESTRIAL  SPHERES.     19 

ing  point  upon  the  celestial  sphere.  This  point  is  called 
the  zenith  of  the  point  on  the  earth.  In  other  words, 
the  zenith  is  defined  by  a  line  passing  through  the  centre 
of  the  earth  to  the  observer,  and  continuing  directly  up- 
ward until  it  meets  the  celestial  sphere.  To  the  observer 
this  line  necessarily  appears  vertical,  because,  wherever  he 
may  be,  he  understands  by  a  vertical  line  one  passing  from 
where  he  stands  toward  the  centre  of  the  earth.  As  the 
earth  revolves,  the  direction  of  this  line  in  relation  to  any 
fixed  diameter  of  the  celestial  sphere  necessarily  varies, 
and  therefore  the  point  in  which  it  cuts  the  celestial  sphere 
or  the  zenith  of  the  observer  varies  also  in  space.  Let  us 
suppose  first  that  the  observer  is  on  the  earth's  equator. 
Then  he  will  see  both  the  north  and  the  south  pole  in  the 
horizon  directly  opposite  each  other.  Looking  upward  he 
will  see  his  zenith  half  way  between  the  poles.  Then,  as 
the  earth  revolves  on  its  axis,  his  zenith  will  describe  a 
great  circle  around  the  celestial  sphere,  every  point  of 
which  will  be  equally  distant  from  the  two  poles.  If  we 
imagine  an  infinitely  long  pencil  reaching  from  any  point 
of  the  earth's  equator  vertically  up  to  the  stars,  we  may 
conceive  that  its  point  marks  out  an  equator  among  them. 
A  complete  revolution  of  the  earth  brings  it  back  to  the 
place  from  which  it  started,  and  thus  completes  the  circle. 
The  imaginary  circle  thus  described  in  the  heavens  is 
called  the  celestial  equator.  The  relation  which  it  bears 
to  the  terrestrial  equator  is  that  every  point  of  it  is  above  a 
corresponding  point  of  the  latter.  The  two  equators  lie 
in  the  same  plane,  passing  through  the  centre  of  the 
earth,  which  plane  is  called  the  plane  of  the  equator,  and 
belongs  to  both  the  celestial  and  terrestrial  spheres. 

Now  suppose  that  the  observer  passes  from  the  equator 
to  45°  of  north  latitude.  His  horizon  having  changed  by 
45°,  the  north  pole  will  now  be  45°  above  the  horizon, 
and  45°  from  the  zenith.  Then,  by  the  revolution  of  the 
earth,  his  zenith  will  describe  a  circle  on  the  celestial 


20  ASTRONOMY. 

sphere  which  will  be  everywhere  45°  distant  from  the 
celestial  equator.  This  circle  will  thus  correspond  to  the 
parallel  of  45°  north  upon  the  earth.  If  he  goes  to  lati- 
tude 60°  north,  he  will  see  the  pole  at  an  elevation  of  60°, 
and  his  zenith  will  in  the  same  way  describe  a  circle  which 
will  be  everywhere  60°  from  the  celestial  equator,  and  30° 
from  the  pole.  If  he  passes  to  the  pole,  the  latter  will 
be  directly  over  his  head,  and  his  zenith  will  not  move  at 


FlG.    7.— TERRESTRIAL  AND  CELESTIAL  SPHERES. 

all.  The  celestial  pole  is  simply  the  point  in  which  the 
earth's  axis  of  rotation,  if  continued  out  in  a  straight  line 
of  infinite  length,  would  meet  the  celestial  sphere.  We 
thus  have  a  series  of  circles  on  the  celestial  sphere  corre- 
sponding to  the  parallels  of  latitude  upon  the  earth. 
Unfortunately  the  celestial  element  corresponding  to 
latitude  on  the  earth  is  not  called  by  that  name,  but  by 
that  of  declination.  The  declination  of  a  star  is  its 
distance  north  or  south  from  the  celestial  equator,  pre- 


CELESTIAL  AND   TERRESTRIAL  MERIDIANS.       21 


FIG.  8. 


cisely  as  latitude  on  the  earth  is  distance  from  the  earth's 
equator. 

Let  L  be  a  place  on  the  earth,  PEp  Q,  Pp  being  the  earth's  axis, 
and  E  Q  its  equator.  Z  is  the 
zenith  and  H  R  the  horizon  of  L. 
L  0  Q  is  the  latitude  of  L  accord- 
ing to  ordinary  geographical  de- 
finitions :  i.e.,  it  is  its  angular  dis- 
tance from  the  equator. 

Prolong  0  P  indefinitely  to  P*t 
and  draw  L  P'  parallel  to  it.  To 
an  observer  at  L  the  elevated  pole 
of  the  heavens  will  be  seen  along 
the  line  L  P"f  because  at  an  in- 
finite distance  the  distance  P  P' 
will  appear  like  a  point.  H  L  Z— 
POQ  and  ZLP"=Z  OP',  hence 
P'L H—LOq— that  is,  the  eleva- 
tion of  the  pole  above  the  celestial 
horizon  is  equal  to  the  latitude  of  the 
place.  Referring  to  Fig.  9,  it  can  at 
once  be  seen  that  the  latitude  of  a 
place  on  the  earth's  surface  is  equal 
to  the  declination  of  the  zenith  of  that 
place,  since  the  declination  of  the  zenith  is  equal  to  the  altitude  of 
the  elevated  pole. 

We  have  next  to  consider  the  correspondence  between 
the  celestial  and  terrestrial  meridians.  A  terrestrial  me- 
ridian is  an  imaginary  line  drawn  along  the  earth's  surface 
in  a  north  and  south  direction  from  one  pole  to  the  other. 
These  meridians  diverge  from  one  pole  in  every  direc- 
tion, and  meet  at  the  other  pole.  Sometimes  they  are 
called  by  the  names  of  places  they  pass  through,  as  the 
meridian  of  Greenwich,  or  the  meridian  of  Washington. 
Each  meridian  may  be  considered  as  the  intersection  with 
the  earth's  surface  of  a  plane  passing  through  the  axis  of 
the  earth,  and  therefore  through  both  poles.  Such  a 
plane  will  cut  the  earth  into  two  equal  hemispheres,  and 
will  of  course  be  vertical  with  the  earth's  surface  along 
every  part  of  its  line  of  intersection.  This  plane  is  called 
the  plane  of  the  meridian  ;  and  by  continuing  it  out  to 
the  celestial  sphere,  we  should  have  a  celestial  meridian 
corresponding  to  each  terrestrial  one,  precisely  as  we  have 


22  ASTRONOMY. 

circles  of  declination  corresponding  to  parallels  of  latitude 
on  the  earth.  But  owing  to  the  rotation  of  the  earth,  the 
circle  in  which  the  plane  of  the  meridian  of  any  place  in- 
tersects the  celestial  sphere  will  be  continually  moving 
among  the  stars,  so  that  there  is  no  such  permanent  cor- 
respondence as  in  the  case  of  the  declinations.  This 
does  not  prevent  us  from  conceiving  imaginary  meridians 
passing  from  one  pole  of  the  heavens  to  the  other  pre- 
cisely as  the  meridians  on  the  earth  do,  only  these  me- 
ridians will  be  apparently  in  motion,  owing  to  the  rotation 
of  the  earth.  We  may,  in  fact,  conceive  of  two  sets  of 
meridians — one  really  at  rest  among  the  stars,  but  appa- 
rently moving  from  east  to  west  around  the  pole  as  the 
stars  do,  and  the  other  the  terrestrial  meridians  continued 
to  the  celestial  sphere,  apparently  at  rest,  but  really  in 
motion  from  west  to  east.  The  relations  of  these  me- 
ridians will  be  best  understood  when  we  explain  the  in- 
struments and  methods  by  which  they  are  fixed,  and  by 
which  the  positions  of  the  stars  in  the  heavens  are  deter- 
mined. At  present  we  will  confine  ourselves  to  the  con- 
sideration of  the  celestial  meridians. 

The  reader  will  understand  that  these  meridians  pass 
from  one  pole  of  the  celestial  sphere  to  the  other,  pre- 
cisely as  on  the  globe  terrestrial  meridians  pass  from  one 
pole  to  the  other,  and  that  being  fixed  among  the  stars, 
they  appear  to  turn  around  the  pole  as  the  stars  appear  to 
do.  As  on  the  earth  differences  of  longitude  between 
different  places  are  fixed  by  the  differences  between  the 
meridians  of  the  two  places,  so  in  the  heavens  what  cor- 
responds to  longitude  is  fixed  by  the  difference  between 
the  celestial  meridians.  This  co-ordinate  is,  however,  in 
the  heavens  not  called  longitude,  but  right  ascension. 
Let  the  student  very  thoroughly  impress  upon  his  mind 
this  term — right  ascension — which  is  longitude  on  the 
celestial  sphere,  and  also  the  term  we  have  before  spoken 
of — declination — which  is  latitude  on  the  celestial  sphere. 

In  order  to  fix  the  right  ascension  of  a  heavenly  body, 


RIGHT  ASCENSION.  23 

we  must  have  a  first  meridian  to  count  from,  precisely  as 
on  the  earth  we  count  longitudes  from  the  meridian  of 
Greenwich  or  of  Washington.  It  is  indifferent  what  me- 
ridian we  take  as  the  first  one  ;  but  it  is  customary  to 
adopt  the  meridian  of  the  vernal  equinox.  What  the  ver- 
nal equinox  is  will  he  described  hereafter  :  for  our  pres- 
ent purposes,  nothing  more  is  necessary  than  to  under- 
stand that  a  certain  meridian  is  arbitrarily  taken.  If  now 
we  wish  to  fix  the  right  ascension  of  a  star,  we  have  only 
to  imagine  a  meridian  passing  through  it,  and  to  deter- 
mine the  angle  which  this  meridian  makes  with  the  meri- 
dian of  the  vernal  equinox,  as  measured  from  west  to  east 
on  the  equator.  That  angle  will  be  the  right  ascension  of 
a  star.  As  already  indicated,  the  declination  of  a  star 
will  be  its  angular  distance  from  the  equator  measured  on 
this  meridian.  Thus,  the  right  ascension  and  declination 
of  a  star  fix  its  apparent  position  on  the  celestial  sphere, 
precisely  as  latitude  and  longitude  fix  the  position  of  a 
point  on  the  surface  of  the  earth. 

To  give  precision  to  the  ideas,  we  present  a  brief  con- 
densation of  this  subject,  with  additional  definitions. 

Let  P  ZRN represent  the  celestial  sphere  of  an  ob- 
server in  the  northern  hemisphere,  0  being  the  position 
of  the  earth.  P  p  is  the  axis  of  the  celestial  sphere,  or 
the  line  about  which  the  apparent  diurnal  orbits  of  the 
stars  and  the  actual  revolution  of  the  earth  are  performed. 

The  zenith,  Z,  is  the  point  immediately  above,  the 
nadir  n,  the  point  immediately  below  the  observer. 
The  direction  Zn  is  defined  in  practice  by  the  position 
freely  assumed  by  the  plumb  line. 

The  celestial  horizon  is  the  plane  perpendicular  to  the 
line  joining  the  zenith  and  nadir  N E S  W ;  or  it  is  the 
terrestrial  horizon  continued  till  it  meets  the  celestial  sphere. 

The  celestial  horizon  intersects  the  earth  in  the  rational 
horizon,  which  passes  through  the  earth's  centre,  and 
which  is  so  called  in  distinction  to  the  sensible  horizon, 
which  is  the  plane  tangent  to  the  earth's  surface  at  any 


24  ASTRONOMY. 

point.  But,  since  the  earth  itself  is  considered  as  but  a 
point  in  comparison  with  the  celestial  sphere,  the  rational 
and  sensible  horizons  are  considered  as  one  and  the  same 
circle  on  this  sphere. 

The  celestial  poles  are  the  extremities  of  the  axis  of  the 
celestial  sphere  P  p,  the  north  pole  being  that  one  which 
is  above  the  horizon  in  the  latitude  of  New  York,  in  the 
northern  hemisphere. 

The  circles  apparently  described  by  the  stars  in  their 
diurnal  orbits  are  called  parallels  of  declination,  KJV  ; 


FlG.    9. — CIRCLES  OF  THE   SPHERE. 

that  one  whose  plane  passes  through  the  centre  of  the 
sphere  being  the  celestial  equator,  or  the  equinoctial, 
CWD. 

The  celestial  equator  is  then  that  parallel  of  declination 
which  is  a  great  circle  of  the  celestial  sphere. 

The  figure  illustrates  the  phenomena  which  appear  in 
the  heavens  to  an  observer  upon  the  earth.  The  stars 
which  lie  in  the  equator  have  their  diurnal  paths  bisected 
by  the  horizon,  and  are  as  long  above  the  horizon  as  below 


CIRCLES  OF  THE  SPHERE.  25 

it ;  those  whose  distances  from  the  pole  (polar-distances) 
are  greater  than  90°  will  be  a  shorter  time  above  the  ho- 
rizon ;  those  whose  polar-distances  are  less  than  90°  a 
longer  time. 

The  circle  N  K  drawn  around  the  pole  P  as  a  centre 
so  as  to  graze  the  horizon  is  called  the  circle  of  perpetual 
apparition,  because  stars  situated  within  it  never  set. 
The  corresponding  circle  S R  round  the  south  pole  is 
called  the  circle  of  perpetual  disappearance,  because  stars 
within  it  never  rise  above  our  horizon. 

The  great  circle  passing  through  the  zenith  and  the 
pole  is  the  celestial  meridian,  NPZS.  The  meridian 
intersects  the  horizon  in  the  meridian  line,  and  the  points 
N  and  S  are  the  north  and  south  points. 

The  prime  vertical,  E ZW,  is  perpendicular  to  the  meri- 
dian line  and  to  the  horizon :  its  extremities  in  the  hori- 
zon are  the  east  and  west  points. 

The  meridian  plane  is  perpendicular  to  the  equator  and 
to  the  horizon,  and  therefore  to  their  intersection.  Hence 
this  intersection  is  the  east  and  west  line,  which  is  thus 
determined  by  the  intersection  of  the  planes  of  the  equator 
and  of  the  horizon. 

The  altitude  of  a  heavenly  body  is  its  apparent  distance 
above  the  horizon,  expressed  in  degrees,  minutes,  and 
seconds  of  arc.  In  the  zenith  the  altitude  is  90°,  which 
is  the  greatest  possible  altitude. 

If  A  be  any  heavenly  body,  the  angle  Z  P  A  which  the 
circle  P  A  drawn  from  the  pole  to  the  body  makes  with 
the  meridian  is  called  the  hour  angle  of  the  body.  The 
hour  angle  is  the  angle  through  which  the  earth  has  ro- 
tated on  its  axis  since  the  body  was  on  the  meridian.  It 
is  so  called  because  it  measures  the  time  which  has 
elapsed  since  the  passage  of  the  body  over  the  meri- 
dian. 

That  diameter  of  the  earth  which  is  coincident  with  the 
constant  direction  of  the  axis  of  the  celestial  sphere  is  its 
axis,  and  intersects  the  earth  in  its  north  and  south  poles. 


26  ASTRONOMY. 

§  4.  THE  DIURNAL  MOTION  IN  DIFFERENT  LATI- 
TUDES. 

As  we  have  seen,  the  celestial  horizon  of  an  observer 
will  change  its  place  on  the  celestial  sphere  as  the  observer 
travels  from  place  to  place  on  the  surface  of  the  earth. 
If  he  moves  directly  toward  the  north  his  zenith  will  ap- 
proach the  north  pole,  but  as  the  zenith  is  not  a  visible 
point,  the  motion  will  be  naturally  attributed  to  the  pole, 
which  will  seem  to  approach  the  point  overhead.  The 
new  apparent  position  of  the  pole  will  change  the  aspect 
of  the  observer's  sky,  as  the  higher  the  pole  appears  above 
the  horizon  the  greater  the  circle  of  perpetual  apparition, 
and  therefore  the  greater  the  number  of  stars,  which 
never  set. 


FlG.    10. — THE   PARALLEL   SPHERE. 

If  the  observer  is  at  the  north  pole  his  zenith  and  the 
pole  itself  will  coincide  :  half  of  the  stars  only  will  be  vis- 
ible, and  these  will  never  rise  or  set,  but  appear  to  move 
around  in  circles  parallel  to  the  horizon.  The  horizon 
and  equator  will  coincide.  The  meridian  will  be  indeter- 
minate since  Z  and  P  coincide  ;  there  will  be  no  east  and 
west  line,  and  no  direction  but  south.  The  sphere  in  this 
case  is  called  a  parallel  sphere. 


DIURNAL  MOTION  IN  DIFFERENT  LATITUDES.    27 

If  instead  of  travelling  to  the  north  the  observer  should 
go  toward  the  equator,  the  north  pole  would  seem  to  ap- 
proach his  horizon.  When  he  reached  the  equator  both 
poles  would  be  in  the  horizon,  one  north  and  the  other 
south.  All  the  stars  in  succession  would  then  be  visible, 
arid  each  would  be  an  equal  time  above  and  below  the 
horizon. 


FlG.    11. — THE  RIGHT   SPHERE. 

The  sphere  in  this  case  is  called  a  right  sphere,  because 
the  diurnal  motion  is  at  right  angles  to  the  horizon.  If  now 
the  observer  travels  southward  from  the  equator,  the  south 
pole  will  become  elevated  above  his  horizon,  and  in  the 
southern  hemisphere  appearances  will  be  reproduced 
which  we  have  already  described  for  the  northern,  except 
that  the  direction  of  the  motion  will,  in  one  respect,  be 
different.  The  heavenly  bodies  will  still  rise  in  the  east 
and  set  in  the  west,  but  those  near  the  equator  will  pass 
north  of  the  zenith  instead  of  south  of  it,  as  in  our  lati- 
tudes. The  sun,  instead  of  moving  from  left  to  right, 
there  moves  from  right  to  left.  The  bounding  line  be- 
tween the  two  directions  of  motion  is  the  equator,  where 
the  sun  culminates  north  of  the  zenith  from  March  till 
September,  and  south  of  it  from  September  till  March. 

If  the  observer  travels  west  or  east  of  his  first  sta- 
tion, his  zenith  will  still  remain  at  the  same  angular 


28  ASTRONOMY. 

distance  from  the  north  pole  as  before,  and  as  the  phe- 
nomena caused  by  the  earth's  diurnal  motion  at  any 
place  depend  only  upon  the  altitude  of  the  elevated  pole 
at  that  place,  these  will  not  be  changed  except  as  to  the 
times  of  their  occurrence.  A  star  which  appears  to  pass 
through  the  zenith  of  his  first  station  will  also  appear  to 
pass  through  the  zenith  of  the  second  (since  each  star  re- 
mains at  a  constant  angular  distance  from  the  pole),  but 
later  in  time,  since  it  has  to  pass  through  the  zenith  of 
every  place  between  the  two  stations.  The  horizons  of 
the  two  stations  will  intercept  different  portions  of  the 
celestial  sphere  at  any  one  instant,  but  the  earth's  rotation 
will  present  the  same  portions  successively,  and  in  the 
same  order,  at  both. 

§  5.  RELATION  OP  TIME  TO  THE  SPHERE. 

As  in  daily  life  we  measure  time  by  the  revolution  of 
the  hands  of  a  clock,  so,  in  astronomy,  we  measure  it  by 
the  rotation  of  the  earth,  or  the  apparent  revolution  of 
the  celestial  sphere.  Since  the  sphere  seems  to  perform 
one  revolution,  or  360°  in  24  hours,  it  follows  that  it 
moves  through  15°  in  one  hour,  1°  in  4  minutes,  15'  in 
one  minute  of  time,  and  15"  in  one  second  of  time. 

The  hour  angle  of  a  heavenly  body  counted  toward  the 
west  (see  definition,  p.  25)  being  the  angle  through  which 
the  sphere  has  revolved  since  the  passage  of  the  body  over 
the  meridian,  it  follows  that  the  time  which  has  elapsed 
since  that  passage  may  be  found  by  dividing  the  hour 
angle,  expressed  in  degrees,  minutes,  and  seconds  of  arc, 
by  15,  when  the  result  will  be  the  required  interval  ex- 
pressed in  hours,  minutes,  and  seconds  of  time.  If  we 
know  the  time  at  which  the  body  passed  the  meridian, 
and  add  this  interval  to  it,  we  shall  have  the  time  corre- 
sponding to  the  hour  angle.  If  we  call  it  noon  when 
the  sun  passes  the  meridian,  the  hour  angle  of  the  sun 
at  any  moment,  divided  by  15,  gives  the  time  since  noon. 
Mean  solar  time  is  our  ordinary  time  measured  by  the 


SIDEREAL   TIME.  29 

sun,  after  allowing  for  certain  inequalities  hereafter  de- 
scribed. 

Here,  however,  an  important  remark  is  to  be  made. 
Really  the  earth  does  not  revolve  on  its  axis  in  24  of  the 
hours  used  in  ordinary  life,  but  in  about  4  minutes  less  than 
24  hours  (more  exactly  in  23  hours  56  minutes  4.09  seconds.) 
If  we  note  the  exact  time  at  which  a  star  crosses  the  meri- 
dian, or  rises  or  sets,  or  disappears  behind  a  chimney  or  other 
terrestrial  object  on  one  night,  we  shall  find  it  to  do  the 
same  thing  3  minutes  56  seconds  earlier  on  the  night  follow- 
ing, an  acceleration  which,  continued  every  day,  amounts  to 
a  whole  day  in  a  year.  The  theory  of  this  acceleration 
will  be  explained  hereafter  as  arising  from  the  annual  revo- 
lution of  the  earth  around  the  sun  ;  at  present  we  are 
concerned  only  with  the  fact.  As  a  consequence  of  this 
fact,  the  starry  sphere  seems  to  revolve  rather  more  than 
15°  in  an  hour,  and  the  relation  between  the  time  and  the 
arc  through  which  the  earth  really  turns,  or  the  sphere 
seems  to  turn,  becomes  complex.  To  avoid  this  complex- 
ity, astronomers  introduce  a  modified  measure  of  time, 
known  as  sidereal  time. 

Sidereal  Time. — The  sidereal  day  is  measured,  not  by 
the  interval  between  two  transits  of  the  sun  over  the  meri- 
dian, but  by  that  between  two  transits  of  the  same  star. 
This  day  is  supposed  to  commence  at  the  moment  of  tran- 
sit of  the  vernal  equinox,  or  the  meridian  from  which  right 
ascensions  are  reckoned  (a  point  among  the  stars  to  be  here- 
after defined),  and  is  about  3  minutes  56  seconds  shorter 
than  the  solar  or  common  day.  It  is,  however,  divided  into 
24  sidereal  hours,  and  the  sidereal  hour  is  subdivided  into 
sidereal  minutes  and  seconds  exactly  like  the  common 
hours.  A  simple  calculation  will  show  that  the  sidereal 
hour  is  nearly  10  seconds  shorter  than  the  solar  hour,  and, 
in  general,  each  unit  of  sidereal  time  is  3  6  6*  2  5  part  short- 
er than  the  corresponding  unit  of  solar  time.  A  sidereal 
clock  is  so  constructed  as  to  gain  on  the  common  clock  at 
this  rate — that  is,  it  gains  about  one  second  in  six  minutes, 


30  ASTRONOMY, 

ten  seconds  in  an  hour,  3  minutes  56  seconds  in  a  day, 
two  hours  in  a  month,  and  24  hours,  or  one  day,  in  a  year. 
The  hours  of  the  sidereal  day  are  counted  forward  from  0 
to  24,  instead  of  being  divided  into  two  groups  of  12  each, 
as  in  our  civil  reckoning  of  time.  The  face  of  the  sidereal 
clock  is  divided  into  24  hours,  and  the  hour  hand 
makes  one  revolution  in  this  period  instead  of  two.  The 
minutes  and  seconds  are  each  counted  forward  from  0  to 
60,  as  in  the  common  clock.  The  hands  are  set  so  as  to 
mark  Oh  Ora  0s  at  the  moment  when  the  vernal  equinox 
passes  the  meridian  of  the  observer.  Thus,  the  sidereal 
time  at  any  moment  is  simply  the  interval  in  hours,  min- 
utes, and  seconds  which  lias  elapsed  since  the  vernal  equi- 
nox was  on  the  meridian.  By  multiplying  this  time  by 
15,  we  have  the  number  of  degrees,  minutes,  and  seconds 
through  which  the  earth  has  turned  since  the  transit  of 
the  vernal  equinox. 

The  sidereal  time  of  our  common  noon  is  given  in  the 
astronomical  ephemeris  for  every  day  of  the  year.  It  can 
be  found  within  ten  or  twelve  minutes  at  any  time  by  re- 
membering that  on  March  22d  it  is  sidereal  0  hours  about 
noon,  on  April  22d  it  is  about  2  hours  sidereal  time  at 
noon,  and  so  on  through  the  year.  Thus,  by  adding  two 
hours  for  each  month,  and  4  minutes  for  each  day  after 
the  22d  day  last  preceding,  we  have  the  sidereal  time  at 
the  noon  we  require.  Adding  to  it  the  number  of  hours 
since  noon,  and  one  minute  more  for  ever  fourth  of  a  day 
on  account  of  the  constant  gain  of  the  clock,  we  have  the 
sidereal  time  at  any  moment. 

Example. — Find  the  sidereal  time  on  July  4th,  1881,  at 
4  o'clock  A.M.  We  have  : 

h        m 

June  22d,  3  months  after  March  22d  ;  to  be  X  2,     6     0 
July  3d,  12  days  after  June  22d  ;  X  4,  0   48 

4  A.M.,  16  hours  after  noon,  nearly  f  of  a  day,        16      3 

22    51 
This  result  is  within  a  minute  of  the  truth. 


SIDEREAL   TIME.  31 

The  reader  now  understands  that  a  sidereal  clock  is  one 
which  keeps  time,  not  by  the  apparent  diurnal  motion  of 
the  sun,  but  by  that  of  the  stars.  Consequently,  the  as- 
tronomer, by  looking  at  his  clock,  always  knows  the 
positions  of  the  stars  relatively  to  his  meridian.  We  have 
now  to  show  how  he  finds  the  right  ascension  of  the  stars 
by  his  sidereal  clock.  This  is  done  by  means  of  the  meri- 
dian transit  instrument,  of  which  we  shall  here  explain  the 
first  principles  of  construction,  reserving  a  full  description 
for  the  chapter  on  instruments.  It  consists  essentially  of 
a  small  telescope  turning  on  an  axis,  which  is  fixed  in  an 
east  and  west  line.  With  the  axis  thus  fixed,  the  tele- 
scope can  turn  only  in  the  plane  of  the  meridian.  When 
the  observer  looks  into  it,  he  will  see  the  apparent 
diurnal  motion  of  any  star  at  which  it  may  point,  and  this 
motion  will  be  magnified  in  the  ratio  of  the  magnifying 
power  of  the  telescope.  With  a  high  power  it  will  there- 
fore appear  very  rapid.  When  the  star  is  exactly  on  the 
meridian  it  will  appear  in  the  middle  of  the  field  of  view 
of  the  telescope,  and,  by  means  of  apparatus  to  be  here- 
after described,  the  moment  of  crossing  can  be  deter- 
mined within  a  small  fraction  of  a  second. 

Suppose  now  that  the  observer  has  his  clock  so  set  that 
it  marks  0  hours  0  minutes  0  seconds  at  the  moment 
that  the  vernal  equinox  crosses  his  meridian,  and  so  regu- 
lated that  when  the  equinox  again  reaches  the  meridian  on 
the  day  following  the  hour  hand  will  have  made  one  revo- 
lution through  the  24  hours,  and  come  back  to  0  hours 
again.  Then,  to  find  the  right  ascension  of  any  star  or 
other  heavenly  body,  he  watches  when  it  is  about  to  reach 
the  meridian  ;  then  directs  the  transit  instrument  at  the 
point  where  it  is  about  to  cross,  and  notes  the  exact  time, 
in  hours,  minutes,  and  seconds,  at  which  the  star  crosses 
the  middle  of  the  field  of  his  transit.  Multiplying  this 
time  by  15,  he  has  the  right  ascension  of  the  star  in  de- 
grees, minutes,  and  seconds.  In  order  to  avoid  the  trouble 
of  this  multiplication,  it  is  now  customary  to  express  the 


32  ASTRONOMY. 

right  ascensions  of  the  heavenly  bodies,  not  in  degrees, 
but  in  time.  The  circle  is  divided  into  24  hours,  like 
the  day,  and  these  hours  are  divided  into  minutes  and 
seconds  in  the  usual  way.  Then  the  right  ascension  of 
a  star  is  the  same  as  the  sidereal  time  at  which  it  passes 
the  meridian. 

The  relation  of  arc  to  time,  as  angular  measures,  can  be 
readily  remembered  by  noting  that  a  minute  or  a  second 
of  time  is  fifteen  times  as  great  as  the  corresponding  de- 
nomination in  arc,  while  the  hour  is  15  times  the  degree. 
The  minute  and  second  of  time  are  denoted  by  the  initial 
letter  of  their  names.  So  we  have  : 

lh  =15°  1°  =4m 

lm  =  15/  1'  =48 

1s  =15"  1"=:09.0666. 


Relation  of  Time  and  Longitude. — Considering  our  civil 
time  as  depending  on  the  sun,  it  will  be  seen  that  it  is 
noon  at  any  and  every  place  on  the  earth  when  the  sun 
crosses  the  meridian  of  that  place,  or,  to  speak  with  more 
precision,  when  the  meridian  of  the  places  passes  under 
the  sun.  In  the  lapse  of  24  hours,  the  rotation  of  the 
earth  on  its  axis  brings  all  its  meridians  under  the  sun  in 
succession,  or,  which  is  the  same  thing,  the  sun  appears  to 
pass  in  succession  all  the  meridians  of  the  earth.  Hence, 
noon  continually  travels  westward  at  the  rate  of  15°  in  an 
hour,  making  the  circuit  of  the  earth  in  24  hours.  The 
difference  between  the  time  of  day,  or  local  time  as  it  is 
called,  at  any  two  places,  will  be  in  proportion  to  the  differ- 
ence of  longitude,  amounting  to  one  hour  for  every  15 
degrees  of  longitude,  four  minutes  for  every  degree,  and 
so  on.  Vice  versa,  if  at  the  same  real  moment  of  time 
we  can  determine  the  local  times  at  two  different  places, 
the  difference  of  these  times,  multiplied  by  15,  will  give 
the  difference  of  longitude. 


CHANGE  OF  DAT.  33 

The  longitudes  of  places  are  determined  astronomically 
on  this  principle.  Astronomers  are,  however,  in  the 
habit  of  expressing  the  longitude  of  places  on  the  earth 
like  the  right  ascensions  of  the  heavenly  bodies,  not  in 
degrees,  but  in  hours.  For  instance,  instead  of  saying 
that  Washington  is  77°  3'  west  of  Greenwich,  we  com- 
monly say  that  it  is  5  hours  8  minutes  12  seconds  west, 
meaning  that  when  it  is  noon  at  Washington  it  is  5  hours 
8  minutes  12  seconds  after  noon  at  Greenwich.  This 
course  is  adopted  to  prevent  the  trouble  and  confusion 
which  might  arise  from  constantly  having  to  change  hours 
into  degrees,  and  the  reverse. 

A  question  frequently  asked  in  this  connection  is, 
Where  does  the  day  change  ?  It  is,  we  will  suppose,  Sun- 
day noon  at  Washington.  That  noon  travels  all  the  way 
round  the  earth,  and  when  it  gets  back  to  Washington 
again  it  is  Monday.  Where  or  when  did  it  change  from 
Sunday  to  Monday  ?  We  answer,  wherever  people  choose 
to  make  the  change.  Navigators  make  the  change 
occur  in  longitude  180°  from  Greenwich.  As  this  meri- 
dian lies  in  the  Pacific  Ocean,  and  scarcely  meets  any  land 
through  its  course,  it  is  very  convenient  for  this  purpose. 
If  its  use  were  universal,  the  day  in  question  would  be 
Sunday  to  all  the  inhabitants  east  of  this  line,  and  Mon- 
day to  every  one  west  of  it.  But  in  practice  there  have 
been  some  deviations.  As  a  general  rule,  on  those  islands 
of  the  Pacific  which  are  settled  by  men  travelling  east, 
the  day  would  at  first  be  called  Monday,  even  though 
they  might  cross  the  meridian  of  180°.  Indeed  the  Eus- 
sian  settlers  carried  their  count  into  Alaska,  so  that  when 
our  people  took  possession  of  that  territory  they  found 
that  the  inhabitants  called  the  day  Monday  when  they 
themselves  called  it  Sunday.  These  deviations  have,  how- 
ever, almost  entirely  disappeared,  and  with  few  exceptions 
the  day  is  changed  by  common  consent  in  longitude  180° 
from  Greenwich. 


34  ASTRONOMY. 

§  6.  DETERMINATIONS  OP  TERRESTRIAL    LONGI- 
TUDES. 

We  have  remarked  that,  owing  to  the  rotation  of  the  earth, 
there  is  no  such  fixed  correspondence  between  meridians  on 
the  earth  and  among  the  stars  as  there  is  between  latitude  on 
the  earth  and  declination  in  the  heavens.  The  observer 
can  always  determine  his  latitude  by  finding  the  declination 
of  his  zenith,  but  he  cannot  find  his  longitude  from  the 
right  ascension  of  his  zenith  with  the  same  facility,  be- 
cause that  right  ascension  is  constantly  changing.  To  deter- 
mine the  longitude  of  a  place,  the  element  of  time  as  mea- 
sured by  the  diurnal  motion  of  the  earth  necessarily  comes 
in.  Let  us  once  more  consider  the  plane  of  the  meridian 
of  a  place  extended  out  to  the  celestial  sphere  so  as  to 
mark  out  on  the  latter  the  celestial  meridian  of  the  place. 
Consider  two  such  places,  Washington  and  San  Francisco 
for  example  ;  then  there  will  be  two  such  celestial  meri- 
dians cutting  the  celestial  sphere  so  as  to  make  an  angle  of 
about  forty-five  degrees  with  each  other  in  this  case.  Let 
the  observer  imagine  himself  at  San  Francisco.  Then  he 
may  conceive  the  meridian  of  Washington  to  be  visible 
on  the  celestial  sphere,  and  to  extend  from  the  pole  over 
toward  his  south-east  horizon  so  as  to  pass  at  a  distance  of 
about  forty-five  degrees  east  of  his  own  meridian.  It 
would  appear  to  him  to  be  at  rest,  although  really  both 
his  own  meridian  and  that  of  Washington  are  moving  in 
consequence  of  the  earth's  rotation.  Apparently  the  stars 
in  their  course  will  first  pass  the  meridian  of  Washington, 
and  about  three  hours  later  will  pass  his  own  meridian. 
Now  it  is  evident  that  if  he  can  determine  the  interval 
which  the  star  requires  to  pass  from  the  meridian  of  Wash- 
ington to  that  of  his  own  place,  he  will  at  once  have  the 
difference  of  longitude  of  the  two  places  by  simply  turn- 
ing the  interval  in  time  into  degrees  at  the  rate  of  fifteen 
degrees  to  each  hour. 

Essentially  the  same  idea  may  perhaps  be  more  readily 
grasped  by  considering  the  star  as  apparently  passing  over 


LONGITUDE,  35 

the  successive  terrestrial  meridians  on  the  surface  of  the 
earth,  the  earth  being  now  supposed  for  a  moment  to  be 
at  rest.  If  we  imagine  a  straight  line  drawn  from  the 
centre  of  the  earth  to  a  star,  this  line  will  in  the  course  of 
twenty-four  sidereal  hours  apparently  make  a  complete 
revolution,  passing  in  succession  the  meridians  of  all  the 
places  on  the  earth  at  the  rate  of  fifteen  degrees  in  an  hour 
of  sidereal  time.  If,  then,  Washington  and  San  Francisco 
are  forty -five  degrees  apart,  any  one  star,  no  matter  what 
its  declination,  will  require  three  sidereal  hours  to  pass 
from  the  meridian  of  Washington  to  that  of  San  Francisco, 
and  the  sun  will  require  three  solar  Jiours  for  the  same 
passage. 

Whichever  idea  we  adopt,  the  result  will  be  the  same  : 
difference  of  longitude  is  measured  by  the  time  required 
for  a  star  to  apparently  pass  from  the  meridian  of  one 
place  to  that  of  another.  There  is  yet  another  way  of 
defining  what  is  in  effect  the  same  thing.  The  sidereal 
time  of  any  place  at  any  instant  being  the  same  with  the 
right  ascension  of  its  meridian  at  that  instant,  it  follows 
that  at  any  instant  the  sidereal  times  of  the  two  places  will 
differ  by  the  amount  of  the  difference  of  longitude.  For 
instance  :  suppose  that  a  star  in  0  hours  right  ascension  is 
crossing  the  meridian  of  Washington.  Then  it  is  0  hours 
of  local  sidereal  time  at  Washington.  Three  hours  later 
the  star  will  have  reached  the  meridian  of  San  Francisco. 
Then  it  will  be  0  hours  local  sidereal  time  at  San  Fran- 
cisco. Hence  the  difference  of  longitude  of  two  places  is 
measured  by  the  difference  of  their  sidereal  times  at  the 
same  instant  of  absolute  time.  Instead  of  sidereal  times, 
we  may  equally  well  take  mean  times  as  measured  by  the 
sun.  It  being  noon  when  the  sun  crosses  the  meridian  of 
any  place,  and  the  sun  requiring  three  hours  to  pass  from 
the  meridian  of  Washington  to  that  of  San  Francisco,  it 
follows  that  when  it  is  noon  at  San  Francisco  it  is  three 
o'clock  in  the  afternoon  at  Washington.* 

*  The  difference  of  longitude  thus  depends  upon  the  angular  dis- 
tance of  terrestrial  meridians,  and  not  upon  the  motion  of  a  celestial  body, 


36  ASTRONOMY. 

The  whole  problem  of  the  determination  of  terrestrial 
longitudes  is  thus  reduced  to  one  of  these  two  :  either 
to  find  the  moment  of  Greenwich  or  Washington  time 
corresponding  to  some  moment  of  time  at  the  place 
which  is  to  be  determined,  or  to  find  the  time  required 
for  the  sun  or  a  star  to  move  from  the  meridian  of  Green- 
wich or  Washington  to  that  of  the  place.  If  it  were 
possible  to  fire  a  gun  every  day  at  Washington  noon 
which  could  be  heard  in  an  instant  all  over  the  earth, 
then  observers  everywhere,  with  instruments  to  deter- 
mine their  local  time  by  the  sun  or  by  the  stars,  wrould  be 
able  at  once  to  fix  their  longitudes  by  noting  the  hour, 
minute,  and  second  of  local  time  at  which  the  gun  was 
heard.  As  a  matter  of  fact,  the  time  of  Washington  noon 
is  daily  sent  by  telegraph  to  many  telegraph  stations,  and 
an  observer  at  any  such  station  who  knows  his  local  time 
can  get  a  very  close  value  of  his  longitude  by  observing  the 
local  time  of  the  arrival  of  this  signal.  Human  ingenuity 
has  for  several  centuries  been  exercised  in  the  effort  to  in- 
vent some  practical  way  of  accomplishing  the  equivalent 
of  such  a  signal  which  could  be  used  anywhere  on  the 
earth.  The  British  Government  long  had  a  standing  offer 
of  a  reward  of  ten  thousand  pounds  to  any  person  who 
would  discover  a  practical  method  of  determining  the  lon- 
gitude at  sea  with  the  necessary  accuracy.  This  reward 
was  at  length  divided  between  a  mathematician  who  con- 
structed improved  tables  of  the  moon's  motion  and  a 
mechanician  who  invented  an  improved  chronometer. 
Before  the  invention  of  the  telegraph  the  motion  of  the 
moon  and  the  transportation  of  chronometers  afforded 
almost  the  only  practicable  and  widely  extended  methods 
of  solving  the  problem  in  question.  The  invention  of 
the  telegraph  offered  a  third,  far  more  perfect  in  its  appli- 

and  hence  the  longitude  of  a  place  is  the  same  whether  expressed  as  a 
difference  of  two  sidereal  times  or  of  two  solar  times.  The  longitude 
of  Washington  west  from  Greenwich  is  5h  8m  or  77°,  and  this  is,  in  fact, 
the  ratio  of  the  angular  distance  of  the  meridian  of  Washington  from 
that  of  Greenwich  to  860°  or  24h.  It  is  thus  plain  that  the  longitude  is 
the  difference  of  the  simultaneous  local  times,  whether  solar  or  sidereal. 


LONGITUDE  BY  CHRONOMETERS.  3? 

cation,  but  necessarily  limited  to  places  in  telegraphic 
communication  with  each  other. 

Longitude  by  Motion  of  the  Moon. — When  we  de- 
scribe the  motion  of  the  rnoon,  we  shall  see  that  it  moves 
eastward  among  the  stars  at  the  rate  of  about  thirteen  de- 
grees per  day,  more  or  less.  In  other  words,  its  right  as- 
cension is  constantly  increasing  at  the  rate  of  a  degree  in 
something  less  than  two  hours.  If,  then,  its  right  ascension 
can  be  predicted  in  advance  for  each  hour  of  Greenwich 
or  Washington  time,  an  observer  at  any  point  of  the 
earth,  by  noting  the  local  time  at  his  station,  when  the 
moon  lias  any  given  right  ascension,  can  thence  determine 
the  corresponding  moment  of  Greenwich  time  ;  and  hence, 
from  the  difference  of  the  local  times,  the  longitude  of  his 
place.  The  moon  will  thus  serve, the  purpose  of  a  sort  of 
clock  running  on  Greenwich  time,  upon  the  face  of  which 
any  observer  with  the  proper  appliances  can  read  the 
Greenwich  hour.  This  method  of  determining  longitudes 
has  its  difficulties  and  drawbacks.  The  motion  of  the 
moon  is  so  slow  that  a  very  small  change  in  its  right  ascen- 
sion will  produce  a  comparatively  large  one  in  the  Green- 
wich time  deduced  from  it — about  27  times  as  great  an 
error  in  the  deduced  longitudes  as  exists  in  the  determi- 
nation of  the  moon's  right  ascension.  With  such  instru- 
ments as  an  observer  can  easily  carry  from  place  to  place, 
it  is  hardly  possible  to  determine  the  moon's  right  ascen- 
sion within  five  seconds  of  arc  ;  and  an  error  of  this 
amount  will  produce  an  error  of  nine  seconds  in  the 
Greenwich  time,  and  hence  of  two  miles  or  more  in  his 
deduced  longitude.  Besides,  the  mathematical  processes 
of  deducing  from  an  observed  right-ascension  of  the  moon 
the  corresponding  Greenwich  time  are,  under  ordinary 
circumstances,  too  troublesome  and  laborious  to  make  this 
method  of  value  to  the  navigator. 

Transportation  of  Chronometers. — The  transportation 
of  chronometers  affords  a  simple  and  convenient  method 
of  obtaining  the  time  of  the  standard  meridian  at  any 
moment.  The  observer  sets  his  chronometer  as  nearly  as 


38  ASTRONOMY. 

possible  on  Greenwich  or  Washington  time,  and  deter- 
mines its  correction  and  rate.  This  he  can  do  at  any  sta- 
tion of  which  the  longitude  is  correctly  known,  and  at 
which  the  local  time  can  be  determined.  Then,  wherever 
he  travels,  he  can  read  the  time  of  his  standard  meridian 
from  the  face  of  his  chronometer  at  any  moment,  and 
compare  it  with  the  local  time  determined  with  his  transit 
instrument  or  sextant.  The  principal  error  to  which  this 
method  is  subject  arises  from  the  necessary  uncertainty  in 
the  rate  of  even  the  best  chronometers.  This  is  the 
method  almost  universally  used  at  sea  where  the  object  is 
simply  to  get  an  approximate  knowledge  of  the  ship's 
position. 

The  accuracy  can,  however,  be  increased  by  carrying  a 
large  number  of  chronometers,  or  by  repeating  the  de- 
termination a  number  of  times,  and  this  method  is  often 
employed  for  fixing  the  longitudes  of  seaports,  etc. 
Between  the  years  1848  and  1855,  great  numbers  of  chro- 
nometers were  transported  on  the  Cunard  steamers  plying 
between  Boston  and  Liverpool,  to  determine  the  difference 
of  longitude  between  Greenwich  and  the  Cambridge  Ob- 
servatory, Massachusetts.  At  Liverpool  the  chronometers 
were  carefully  compared  with  Greenwich  time  at  a  local 
observatory — that  is,  the  astronomer  at  Liverpool  found 
the  error  of  the  chronometer  on  its  arrival  in  the  ship, 
and  then  again  when  the  ship  was  about  to  sail.  When 
the  chronometer  reached  Boston,  in  like  manner  its  error 
on  Cambridge  time  was  determined,  and  the  determination 
was  repeated  when  the  ship  was  about  to  return.  Having 
a  number  of  such  determinations  made  alternately  on  the 
two  sides  of  the  Atlantic,  the  rates  of  the  chronometers 
could  be  determined  for  each  double  voyage,  and  thus  the 
error  on  Greenwich  time  could  be  calculated  for  the  mo- 
ment of  each  Cambridge  comparison,  and  the  moment  of 
Cambridge  time  for  each  Greenwich  comparison. 

Longitude  by  the  Electric  Telegraph. — As  soon  as  the 
electric  telegraph  was  introduced  it  was  seen  by  American 


LONGITUDE  BY  TELEGRAPH.  39 

astronomers  that  we  here  had  a  method  of  determining 
longitudes  which  for  rapidity  and  convenience  would 
supersede  all  others.  The  first  application  of  this  method 
was  made  in  1844  between  Washington  and  Baltimore, 
under  the  direction  of  the  late  Admiral  Charles  Wilkes, 
U.  S.  N.  During  the  next  two  years  the  method  was  intro- 
duced into  the  Coast  Survey,  and  the  difference  of  longitude 
between  New  York,  Philadelphia,  and  Washington  was 
thus  determined,  and  since  that  time  this  method  has  had 
wide  extension  not  only  in  the  United  States,  but  between 
America  and  Europe,  in  Europe  itself,  in  the  East  and  West 
Indies,  and  South  America.  The  principle  of  the  method 
is  extremely  simple.  Each  place,  of  which  the  difference  of 
time  (or  longitude)  is  to  be  determined,  is  furnished  with  a 
transit  instrument,  a  clock  and  a  chronograph  ;  instruments 
described  in  the  next  chapter.  Each  clock  is  placed  in 
galvanic  communication  not  only  with  its  own  chronograph, 
but  if  necessary  is  so  connected  with  the  telegraph  wires 
that  it  can  record  its  own  beat  upon  a  chronograph  at  the 
other  station.  The  observer,  looking  into  the  telescope 
and  noting  the  crossing  of  the  stars  over  the  meridian, 
can,  by  his  signals,  record  the  instant  of  transit  both  on  his 
own  chronograph  and  on  that  of  the  other  station.  The 
plan  of  making  a  determination  between  Philadelphia  and 
Washington,  for  instance,  was  essentially  this  :  When 
some  previously  selected  star  reached  the  meridian  at  Phil- 
adelphia, the  observer  pointed  his  transit  upon  it,  and  as 
it  crossed  the  wires,  recorded  the  signal  of  time  not  only 
on  his  own  chronograph,  but  on  that  at  Washington. 
About  eight  minutes  afterward  the  star  reached  the 
meridian  at  Washington,  and  there  the  observer  recorded 
its  transit  both  on  his  own  chronograph  and  on  that  at 
Philadelphia.  The  interval  between  the  transit  over  the 
two  places,  as  measured  by  either  sidereal  clock,  at  once 
gave  the  difference  of  longitude.  If  the  record  was  in- 
stantaneous at  the  two  stations,  this  interval  ought  to  be 
the  same,  whether  read  off  the  Philadelphia  or  the  Wash- 


40  ASTRONOMY. 

ington  chronograph.  It  was  found,  however,  that  there 
was  a  difference  of  a  small  fraction  of  a  second,  arising 
from  the  fact  that  electricity  required  an  interval  of  time, 
minute  but  yet  appreciable,  to  pass  between  the  two 
cities.  The  Philadelphia  record  was  a  little  too  late  in 
being  recorded  at  Washington,  and  the  Washington  one  a 
little  too  late  in  being  recorded  at  Philadelphia.  We 
may  illustrate  this  by  an  example  as  follows  : 

Suppose  E  to  be  a  station  one  degree  of  longitude  east 
of  another  station,  W  ;  and  that  at  each  station  there  is  a 
clock  exactly  regulated  to  the  time  of  its  own  place,  in 
which  case  the  clock  at  E  will  be  of  course  four  minutes 
fast  of  the  clock  at  W  ;  let  us  also  suppose  that  a  signal 
takes  a  quarter  of  a  second  to  pass  from  one  station  to  the 
other  : 

Then  if  the  observer  at  E  sends  a  signal  to  W  at  exactly 

noon  by  his  clock 12h    Om  0s. 00 

It  will  be  received  at  W  at . .  llh  56m  0*.25 


Showing  an  apparent  difference  of  time  of 3m  599.75 

Then  if  the  observer  at  W  sends  a  signal  at  noon  by  his 

clock 12h  Om  08.00 

It  will  be  received  at  E  at  .  12h  4m  Os.25 


Showing  an  apparent  difference  of  time  of 4m  0s. 25 

One  half  the  sum  of  these  differences  is  four  minutes, 
which  is  exactly  the  difference  of  time,  or  one  degree  of 
longitude  ;  and  one  half  their  difference  is  twenty-five 
hundredths  of  a  second,  the  time  taken  by  the  electric  im- 
pulse to  traverse  the  wire  and  telegraph  instruments. 

This  is  technically  called  the  "  wave  and  armature 
time." 

We  have  seen  that  if  a  signal  could  be  made  at  Wash- 
ington noon,  and  observed  by  an  observer  anywhere  sit- 
uated who  knew  the  local  time  of  his  station,  his  longi- 
tude would  thus  become  known.  This  principle  is  often 
employed  in  methods  of  determining  longitude  other  than 
those  named.  For  example,  the  instant  of  the  beginning 


THEORY  OF  THE  SPHERE.  41 

and  ending  of  an  eclipse  of  the  sun  (by  the  moon)  is  a 
perfectly  definite  phenomenon.  If  this  is  observed  by 
two  observers,  and  these  instants  noted  by  each  in  the 
local  time  of  his  station,  then  the  difference  of  these 
local  times  (subject  to  small  corrections,  due  to  parallax, 
etc.)  will  be  the  difference  of  longitude  of  the  two  sta- 
tions. 

The  satellites  of  Jupiter  suffer  eclipses  frequently,  and 
the  Greenwich  and  Washington  times  of  these  phenomena 
are  computed  and  set  down  in  the  Nautical  Almanac.  Ob- 
servations of  these  at  any  station  will  also  give  the  differ- 
ence of  longitude  between  this  station  and  Greenwich  or 
Washington.  As,  however,  they  require  a  larger  tele- 
scope and  a  higher  magnifying  power  than  can  be  used  at 
sea,  this  method  is  not  a  practical  one  for  navigators. 


§  7.  MATHEMATICAL  THEORY  OP  THE  CELESTIAL 
SPHERE. 

In  this  explanation  of  the  mathematical  theory  of  the  relations  of 
the  heavenly  bodies  to  circles  on  the  sphere,  an  acquaintance  with 
spherical  trigonometry  on  the  part  of  the  reader  is  necessarily  pre- 
supposed. The  general  method  by  which  the  position  of  a  point  on 
the  sphere  is  referred  to  fixed  points  or  circles  is  as  follows : 

A  fundamental  great  circle  E  V  Q,  Fig.  12  is  taken  as  a  basis, 
and  the  first  co-ordinate  *  of  the  body  is  its  angular  distance  from 
this  circle.  When  the  earth's  equator  is  taken  as  the  fundamental 
circle,  this  distance  is  on  the  earth's  surface  called  Latitude  ;  on  the 
celestial  sphere  the  corresponding  distance  is  called  Declination.  If 
the  horizon  is  taken  as  the  fundamental  circle  the  distance  is  called 
Altitude.  Altitude  is  therefore  angular  distance  above  the  horizon. 
To  distinguish  between  distances  on  opposite  sides  of  the  circle,  dis- 
tances on  one  side  are  regarded  as  algebraically  positive  quantities, 
and  on  the  other  side  as  negative.  In  the  case  of  the  equator  the 
north  side,  and  in  that  of  the  horizon  the  upper  side,  are  considered 
positive.  Hence,  if  a  body  is  below  the  horizon  its  altitude  is  nega- 
tive, and  the  latitude  of  a  city  south  of  the  earth's  equator  is,  in 
astronomical  language,  considered  as  negative. 

Instead  of  the  co-ordinate  we  have  described,  another  called  zenith 
or  polar  distance  is  frequently  employed.  The  fundamental  circle  is 

*  The  co-ordinates  of  a  body  are  those  measures,  whether  of  angles  or  linex,  which 
define  its  position.  For  instance,  the  geographical  co-ordinates  of  a  city  are  its 
latitude  and  longitude.  To  ftx  a  position  on  a  sphere  or  other  surface,  two  co-ordi- 
nates are  necessary,  while  in  space  three  are  required. 


ASTRONOMY. 


everywhere  90°  from  its  positive  pole,  P.     Hence,  if  A  is  the  position 

of  a  star  or  other  point  on  the 
sphere,  and  we  put 

d,    its    declination    or   altitude, 
=  aA. 

p,  its  polar  or   zenith  distance 
~P  A,  we  shall  have 


or, 


d  +  p  =  90°, 
p=  90°— 6, 


If  the  star  is  south  of  the 
fundamental  circle,  at  B  for  ex- 
ample, 6  being  negative^  will  ex- 
ceed 90°.  This  quantity  p  may 
range  from  zero  at  the  one  pole 
FIG.  12.  to  180°  at  the  other,  and  will  al- 

ways  be  algebraically  positive. 

It  is  on  this  account  to  be  preferred  to  &,  though  less  frequently 
used. 

II.  The  second  co-ordinate  required  to  fix  a  position  on  the  celes- 
tial or  terrestrial  sphere  is  longitude,  right  ascension,  or  azimuth,  ac- 
cording to  the  fundamental  plane  adopted.  It  is  expressed  by  the 
position  of  the  great  circle  or  meridian  P  A  a  F  which  passes 
through  the  position  from  one  pole  to  the  other,  at  right  angles  to 
the  fundamental  circle.  An  arbitrary  point,  V  for  instance,  is  chosen 
on  this  latter  circle,  and  the  longitude  is  the  angle  Va,  from  this 
point  to  the  intersection  of  the  meridian  or  vertical  circle  passing 
through  the  object.  We  may  also  consider  it  as  the  angle  V  P  A 
which  the  circle  passing  through  the  object  makes  with  the  circle 
P  V,  because  this  angle  is  equal 
to  Va.  The  angle  is  commonly 
counted  from  V  toward  the  right, 
and  from  0°  round  to  360°,  so  as 
to  avoid  using  negative  angles. 
If  the  observer  is  stationed  in 
the  centre  of  the  sphere,  with  his 
head  toward  the  positive  pole  P, 
the  positive  direction  should  be 
from  right  to  left  around  the 
sphere.  When  the  horizon  is 
taken  as  the  fundamental  circle 
or  plane,  this  secondary  co-ordi- 
nate is  called  the  azimuth,  and 
should  be  counted  from  the  south 
point  toward  east,  or  from  the  FIG.  13. 

north  point   toward  west,  but  is 

commonly  counted  the  other  way.  It  may  be  defined  as  the  angular 
distance  of  the  vertical  circle  passing  through  the  object  from  the 
south  point  of  the  horizon. 


THEORY  OF  THE  SPHERE.  43 

The  hour  angle  of  a  star  is  measured  by  the  interval  which  has 
elapsed,  or  the  angle  through  which  the  earth  has  revolved  on  its 
axis,  since  the  star  crossed  the  meridian.  In  Fig.  13  Z  being  the 
zenith  and  P  the  pole,  the  angle  Z  P  8  is  the  hour  angle  of  the  star 
8.  This  angle  is  measured  at  the  pole.  If  we  put 

r,  the  sidereal  time, 

<r,  the  right  ascension  of  the  object,  we  shall  have 

Hour  angle,  h  =  T  —  a. 

It  will  be  negative  before  the  object  has  passed  the  meridian,  and 
positive  afterward.  It  differs  from  right  ascension  only  in.  the  point 
from  which  it  is  reckoned,  and  the  direction  which  is  considered 
positive.  The  right  ascension  is  measured  toward  the  east  from  a 
point  (the  vernal  equinox)  which  is  fixed  among  the  stars,  while  the 
hour  angle  is  measured  toward  the  west  from  the  meridian  of  the 
observer,  which  meridian  is  constantly  in  motion,  owing  to  the 
earth's  rotation. 

We  have  next  to  show  the  trigonometrical  relations  which  subsist 
between  the  hour  angle,  declination,  altitude,  and  azimuth.  Let 


FIG.  14. 


Fig.   14  be  a  view  of  the  celestial  hemisphere  which  is  above  the 
horizon,  as  seen  from  the  east.     We  then  have  : 

HER  TF,  the  horizon. 

P,  the  pole. 

Z,  the  zenith  of  the  observer.  ^ 

H  M  Z  P  R,  the  meridian  of  the  observer. 

P  R,  the  latitude  of  the  observer,  which  call  (j>. 

Z  P,  =  90°  —  0,  the  co-latitude. 

P  8,  the  north  polar  distance  of  the  star  =  90"  —  declination. 

T  8,  its  altitude,  which  call  a. 

Z  8,  its  zenith  distance  =  90°  —  a. 

M  Z  #,  its  azimuth,  =  180°  —  angle  S  Z  P. 

Z  P  S,  its  hour  angle,  which  call  h. 

The  spherical  triangle  Z  P  8,  of  which  the  angles  are  formed   by 


44  ASTRONOMY. 

the  zenith,  the  pole,  and  the  star,  is  the  fundamental  triangle  of  our 
problem.  The  latter,  as  commonly  solved,  may  be  put  into  two  forms. 

I.  Given  the  latitude  of  the  place,  the  declination    or  polar  dis- 
tance of  the  star,  and  its  hour  angle,  to  find  its  altitude  and  azimuth. 

We  have,  by  spherical  trigonometry,  considering  the  angles  and 
sides  of  the  triangle  Z  P  8 : 

cos  Z  8  =  cos  PZ  cos  P  S  +  sin  P  Z  sin  P  8  cos  P. 
sin  Z  8  cos  Z  =  sin  P  Z  cos  PS  —  cos  P  Z  sin  P  8  cos  P. 
sin  Z  8  sin  Z  =  sin  P  S  sin  P. 

By  the  above  definitions, 

Z  S  —  90°  —  a,  (a  being  the  altitude  of  the  star). 
P  Z  =  90°  —  <£,  0  being  the  latitude  of  the  place). 
P8  =  90°  —  <5,  (6  being  the  declination  of  the  star,  +  when  north). 

P  =  A,  the  hour  angle. 

Z  =  180°  —  z,  (z  being  the  azimuth). 

Making  these  substitutions,  the  equation  becomes : 

sin  a  =  sin  0  sin  <J  +  cos  0  cos  6  cos  h. 
cos  a  cos  z  =  —  cos  {&  sin  <5  +  sin  <j>  cos  <5  cos  h. 
cos  a  sin  z  =  cos  (5  sin  A. 

From  these  equations  sin  a  and  cos  a  may  be  obtained  separately, 
and,  if  the  computation  is  correct,  they  will  give  the  same  value  of  a. 
If  the  altitude  only  is  wanted,  it  may  be  obtained  from  the  first 
equation  alone,  which  may  be  transformed  in  various  ways,  explained 
in  works  on  trigonometry. 

II.  Given  the  latitude  of  the  place,  the  'declination  of  a  star,  and 
its  altitude  above  the  horizon,  to  find  its  hour  angle  and  (if  its  right 
ascension  is  known)  the  sidereal  time  when  it  had  the  given  altitude. 

We  find  from  the  first  of  the  above  equations, 

sin  a  —  sin  6  sin  6 

cos  ti  =  -  -i-3 ; 

cos  tf>  cos  o 

or  we  may  use : 

,  cos  (0  —  d)  —  sin  a 

sm2M  =  I  —  — 7 • 

cos  0  cos  o 

Having  thus  found  h,  we  have 

Sidereal  time  =  h  -f  a, 

a  being  the  star's  right  ascension,  and  the  hour  angle  h  being  changed 
into  time  by  dividing  by  15. 

III.  An  interesting  form  of  this  last  problem  arises  when  we  suj> 
pose  a  =  0,  which  is  the  same  thing  as  supposing  the  star  to  be  in 


ASTRONOMY. 


45 


the  horizon,  and  therefore  to  be  rising  or  setting.  The  value  of  h 
will  then  be  the  hour  angle  at  which  it  rises  or  sets  ;  or  being 
changed  to  time  by  dividing  by  15,  it  will  be  the  interval  of  sidereal 
time  between  its  rising  and  its  passage  over  the  meridian,  or  be- 
tween this  passage  and  its  setting.  This  interval  is  called  the  semi- 
diurnal arc,  and  by  doubling  it 
we  have  the  time  between  the 
rising  and  setting  of  the  star  or 
other  object.  Putting  a  =  0  in 
the  preceding  expression  for  cos 
h  we  find  for  the  semi-diurnal 
arc  A, 

sin  6  sin  d 

cos  h  = 

cos  (j>  cos  6 

=  —  tan  $  tan  6, 

and  the  arc  during  which    the 
star  is  above  the  horizon  is  2  h. 

From   this   formula    may   be 

deduced   at  once   many  of   the    ^ __^^_ 
results  given  in   the   preceding   Fm.  15._UPPER  AND  LOWER  DITJR. 
sections. 

(1).  At  the  poles   0   =  90°, 

tan  ^  =  infinity,  and  therefore  cos  Ji  =  infinity.  But  the  cosine  of 
an  angle  can  never  be  greater  than  unity  ;  there  is  therefore  no  value 
of  h  which  fulfils  the  condition.  Hence,  a  star  at  the  pole  can 
neither  rise  nor  set. 

(2).  At  the  earth's  equator  0  =  0°,  tan  0  =  0,  whence  cos  Ji  =  0, 
h  =  90°,  and  2  Ji  =  180°,  whatever  be  6.  This  being  a  semicircum- 
ferenceall  the  heavenly  bodies  are  half  the  time  above  the  horizon  to 
an  observer  on  the  equator. 

(3).  If  6  =  0°  (that  is,  if  the  star  is  on  the  celestial  equator),  then 
tan  (5  =  0,  and  cos  h  =  0,  h  =  90°,  2  h  =  180°,  so  that  all  stars  on 
the  equator  are  half  the  time  above  the  horizon,  whatever  be  the  lati- 
tude of  the  observer.  Here  we  except  the  pole,  where,  in  this  case, 
tan  q  tan  6  =  a  x  0,  an  indeterminate  quantity.  In  fact,  a  star  on 
the  celestial  equator  will,  at  the  pole  of  the  earth,  seem  to  move  round 
in  the  horizon. 

(4).  The  above  value  of  cos  h  may  be  expressed  in  the  form : 


cos  h  =  — 


tan  d 
cot  <£ 


tan 


tan  (90°  —  4> 


This  shows  that  when  6  lies  outside  the  limits  +  (90°  —  0)  and 
—  (90°  —  0),  cos  Ji  will  lie  without  the  limits  —  1  and  +  1,  and 
there  will  be  no  value  of  h  to  correspond.  Hence,  in  this  case,  the 
stars  neither  rise  nor  set.  These  limits  correspond  to  those  of  per- 
petual apparition  and  perpetual  disappearance. 

(5).  In  the  northern  hemisphere  0  and  tan.  $  are  positive.  Then, 
when  6  is  positive,  cos  Ji  is  negative,  and  h  >  90°,  2  h  >  180".  With 


46  ASTRONOMY. 

negative  3,  cos  h  is  positive,  h  <  90°,  2  h  <  180°.  Hence,  in  north- 
ern latitudes,  a  northern  star  is  more  than  half  of  the  time  above  the 
horizon,  and  a  southern  star  less.  In  the  southern  hemisphere,  0  and 
tan  0  are  negative,  and  the  case  is  reversed. 

(6).  If,  in  the  preceding  case,  the  declination  of  a  body  is  supposed 
constant  and  north,  then  the  greater  we  make  $  the  greater  the  nega- 
tive value  of  cos  h  and  the  greater  h  itself  will  be.  Considering,  in 
succession,  the  cases  of  north  and  south  declination  and  north  and 
south  latitude,  we  readily  see  that  the  farther  we  go  to  the  north  on 
the  earth,  the  longer  bodies  of  north  declination  remain  above  the 
horizon,  and  the  more  quickly  those  of  south  declination  set.  In  the 
southern  hemisphere  the  reverse  is  true.  Thus,  in  the  month  of 
June,  when  the  sun  is  north  of  the  equator,  the  days  are  shortest 
near  the  south  pole,  and  continually  increase  in  length  as  we  go  north. 

EXAMPLES. 

(1).  On  April  9,  1879,  at  Washington,  the  altitude  of  Rigel  above 
the  west  horizon  was  observed  to  be  12°  25'.  Its  position  was: 

Right  ascension  =  5h  8m  44s -27  =  a. 

Declination  =  —  8°  20'  36"   =  <*. 

The  latitude  of  Washington  is  +  38°  53'  39"  =  </>. 

What  was  the  hour  angle  of  the  star,  and  the  sidereal  time  of  ob- 
servation ? 

Ig  sin  a  =        9  •  332478 

Igsin  0  =        9-797879 
Igsin  6  =  —  9-161681 

—  Ig  sin  0  sin  6  =        8-959560 

—  sin  0  sin  6  =        0-091109 
sin«:=        0-215020 

sin  a  —  sin  0  sin  6  =        0-306129 

Igcos0=        9-891151 
Ig  cos  fi  =        9-995379 

Ig  cos  0  cos  6  =        9-886530 
Ig  (sin  a  —  sin  <£  sin  (5,  =        9-485905 


Ig  cos  h  = 

9-599375 

h  = 

66°  34'  33" 

h  - 

h  15   = 

4h  26m  18s.  20 

a  = 

5h    8m44s.27 

sidereal 

time  = 

9h  35m    2S.47 

(2).  Had  the  star  been  observed  at  the  same  altitude  in  the  east, 
what  would  have  been   the  sidereal  time  ? 
Ans.    oc  -  h  =  Oh  42m  269.07. 


DETERMINATION  OF  LATITUDE. 


47 


(3).  At  what  sidereal  time  does  Rigel  rise,  and  at  what  sidereal 
time  does  it  set  in  the  latitude  of  Washington  ? 

—  tg  0  =  -  9-906728 
tg  6  =  —  9-166301 


cos  h  =. 
h  = 


h  -4- 


15  = 

a  = 


9  073029 

3^  12'  19" 
5h  32n,  49s  27 

5h    8'"44S.27 


rises  23h  35"'55\00 
sets  101'  41m  33S.54 

(4).  What  is  the  greatest  altitude  of  Rigel  above  the  horizon  of 
Washington,  and  what  is  its  greatest  depression  below  it  V  Ans. 
Aititude=42  '  45'  45"  ;  depression=59°  26'  57". 

(5).  What  is  the  greatest  altitude  of  a  star  on  the  equator  in  the 
meridian  of  Washington  ?  Ans.  51"  6'  21". 

(6).  The  declination  of  the  pointer  in  the  Great  Bear  which  is 
nearest  the  pole  is  62°  30'  N.,  at  what  altitude  does  it  pass  above 
the  pole  at  Washington,  and  at  what  altitude  does  it  pass  below  it  V 
Ans.  66°  23'  39"  above  the  pole,  and  11°  23'  39'  when  below  it. 

(7).  If  the  declination  of  a  star  is  50"  N.,  what  length  of  sidereal 
time  is  it  above  the  horizon  of  Washington  and  what  length  below  it 
during  its  apparent  diurnal  circuit?  Ans.  Above,  21h  52'"  ;  below, 


§    8.    DETERMINATION    OP    LATITUDES     ON    THE 
EARTH  BY  ASTRONOMICAL  OBSERVATIONS. 

Latitude  from  circumpolar  stars.  —  In  Fig.  16  let  Z  represent  the 
zenith  of  the  place  of  observation,  P  the  pole,  and  HPZ  R  the  me- 
ridian, the  observer  being  at  the 
centre  of  the  sphere.  Suppose 
S  and  8'  to  be  the  two  points 
at  which  a  circumpolar  star 
crosses  the  meridian  in  the  de- 
scription of  its  apparent  diurnal 
orbit.  Then,  since  P  is  midway 
between  S  and  -6'', 

ZS  +  ZS' 

-  --  -----  =  ZP  =  90°  -  0, 


or, 


Z+  Z' 


If,  then,  we  can  measure  the  dis- 
tances Z  and  Z' ,  we  have 

Z+  Z' 


<p  =  90°  — 

21 

which  serves  to  determine 


FIG.  16. 
The  distances  Z  and  Z'  can  be  meas- 


48  ASTRONOMY. 

ured  by  the  meridian  circle  or  the  'sextant — both  of  which  instru- 
ments are  described  in  the  next  chapter — and  the  latitude  is  then 
known.  Z  and  Z'  must  be  freed  from  the  effects  of  refraction.  In 
this  method  no  previous  knowledge  of  the  star's  declination  is  re- 
quired, provided  it  remains  constant  between  the  upper  and  lower 
transit,  which  is  the  case  for  fixed  stars. 

Latitude  by  Circum-zenith  Observations.— If  two  stars 
S  and  8',  whose  declinations  6  and  6'  are  known,  cross  the  meridian, 
one  north  and  the  other  south  of  the  zenith,  at  zenith  distances  Z  8 

and  Z8',  which  call  Z  and  Z',  and 
if  we  have  measured  Z  and  Z\  we 
can  from  such  measures  find  the 
latitude  ;  f or  0  =  6  +  Z  and  <t>  — 
<T  —  Z',  whence 

9  =  $[(<$  + 6') +  (Z-Z')]. 

It  will  be  noted  that  in  this  meth- 
od the  latitude  depends  simply 
upon  the  mean  of  two  declinations 
which  can  be  determined  before- 
hand, and  only  requires  the  differ- 
FL&.  17.  ence  of  zenith  distances  to  be  ac- 

curately measured,  while  the  ab- 
solute values  of  these  are  unknown.  In  this  consists  its  capital  ad- 
vantage. This  is  the  method  invented  by  Capt.  ANDREW  TALCOTT, 
U.S.A.,  and  now  universally  adopted  in  America  in  field  astronomy, 
in  the  practice  of  the  Coast  Survey,  etc. 

Latitude  by  a  Single  Altitude  of  a  Star. — In  the  triangle 
ZP S  (Fig.  14)  the  sides  are  ZP  —  90°  —  <j> ;  PS=  90°  —  <J;  Z 8  = 
Z  •=.  90°  —  a-  ZPS  =  h  =  the  hour  angle.  If  we  can  measure  at 
any  known  sidereal  time  0  the  altitude  a  of  the  star  /8,  and  if  we 
further  know  the  right  ascension,  a,  and  the  declination,  (5,  of  the 
body  (to  be  derived  from  the  Nautical  Almanac  or  a  catalogue  of 
stars),  then  we  have  from  the  triangle 

sin  0  =  sin  a  sin  6  +  cos  a  cos  6  cos  h  ; 
or,  since 

t  =  Q  —  a  ;   sin  <t>  =  sin  a  sin  6  +  cos  a  cos  <5  cos  (0  —  a), 

from  which  we  can  obtain  A.  It  is  to  be  noted  that  in  a  place  whose 
latitude  (0)  is  known,  this  observation  will  determine  0,  the  side- 
real time,  as  explained  in  the  last  section ;  if  the  sun  is  observed, 
t  is  simply  the  solar  time. 

Latitude  by  a  Meridian  Altitude. — If  the  altitude  of  the 
body  is  observed  on  the  meridian  and  south  of  the  zenith,  the  equa- 
tion above  becomes,  since  h  =  0  in  this  case, 

sin  0  =  sin  a  sin  6  -f  cos  a  cos  (5, 
or, 

sin  0  =  cos  (  a  —  6)  . '.  0  =  90°  —  a  +  (5, 

which  is  evidently  the  simplest  method  of  obtaining  0  from  a  meas- 


PARALLAX.  49 

ured  altitude  of  a  body  of  known  declination.  The  last  method  is 
that  commonly  used  at  sea,  the  altitude  being  measured  by  the  sex- 
tant. The  student  can  deduce  the  formula  for  a  northern  altitude. 


§  9.    PARALLAX  AND  SEMIDIAMETEB. 

An  observation  of  the  apparent  position  of  a  heavenly 
body  can  give  only  the  direction  in  which  it  lies  from  the 
station  occupied  by  the  observer  without  any  direct  indi- 
cation of  the  distance.  It  is  evident  that  two  observers 
stationed  in  different  parts  of  the  earth  will  not  see  such 
a  body  in  the  same  direction.  In  Fig.  18,  let  $'  be  a  sta- 


FlQ.    18.— PARALLAX. 

tion  on  the  earth,  P  a  planet,  Z'  the  zenith  of  $',  and  the 
outer  arc  a  part  of  the  celestial  sphere.  An  observation 
of  the  apparent  right  ascension  and  declination  of  P  taken 
from  the  station  S'  will  give  us  an  apparent  position  P'. 
A  similar  observation  at  Sff  will  give  an  apparent  position 
P",  while  if  seen  from  the  centre  of  the  earth  the  appar- 
ent position  would  be  Pr  The  angles  P'  P  P ,  and 
P"  P  Pt,  which  represent  the  differences  of  direction,  are 
called  parallaxes.  It  is  clear  that  the  parallax  of  a  body 
depends  upon  its  distance  from  the  earth,  being  greater 
the  nearer  it  is  to  the  earth. 

The  word  parallax  having  several  distinct  applications, 
we  shall  give  them  in  order,  commencing  with  the  most 
general  signification. 


50  ASTRONOMY. 

(1.)  In  its  most  general  acceptation,  parallax  is  the  differ- 
ence between  the  directions  of  a  body  as  seen  from  two 
different  standpoints.  This  difference  is  evidently  equal 
to  the  angle  made  between  two  lines,  one  drawn  from  each 
point  of  observation  to  the  body.  Thus  in  Fig.  18  the 
difference  between  the  direction  of  the  body  P  as  seen 
from  C  and  from  S'  is  equal  to  the  angle  P'  P  P (,  and  this 
again  is  equal  to  its  opposite  angle  'S  P  C. '  This  angle  is, 
however,  the  angle  between  the  two  points  C  and  8'  as 
seen  from  P  :  we  may  therefore  refer  this  most  general 
definition  of  parallax  to  the  body  itself,  and  define  parallax 
as  the  angle  subtended  by  the  line  between  two  stations  as 
seen  from  a  heavenly  body. 

(2.)  In  a  more  restricted  sense,  one  of  the  two  stations  is 
supposed  to  be  some  centre  of  position  from  which  we 
imagine  the  body  to  be  viewed,  and  the  parallax  is  the 
difference  between  the  direction  of  the  body  from  this 
centre  and  its  direction  from  some  other  point.  Thus 
the  parallax  of  which  we  have  just  spoken  is  the  differ- 
ence between  the  direction  of  the  body  as  seen  from  the 
centre  of  the  earth  C  and  from  a  point  on  its  surface  as  &. 
If  the  observer  at  any  station  on  the  earth  determines 
the  exact  direction  of  a  body,  the  parallax  of  which  we 
speak  is  the  correction  to  be  applied  to  that  direction  in 
order  to  reduce  it  to  what  it  would  have  been  had  the  ob- 
servation been  made  at  the  centre  of  the  earth.  Obser- 
vations made  at  different  points  on  the  earth's  surface  are 
compared  by  reducing  them  all  to  the  centre  of  the  earth. 

We  may  also  suppose  the  point  C  to  be  the  sun  and  the 
circle  S'  S"  to  be  the  earth's  orbit  around  it.  The  paral- 
lax will  then  be  the  difference  between  the  directions  of 
the  body  as  seen  from  the  earth  and  from  the  sun.  This 
is  termed  the  annual  parallax,  because,  owing  to  the  an- 
nual revolution  of  the  earth,  it  goes  through  its  period 
in  a  year,  always  supposing  the  body  observed  to  be  at 
rest. 

(3.)  A   yet  more  restricted  parallax  is  the  horizontal 


PARALLAX.  51 

parallax  of  a  heavenly  body.  The  parallax  first  described 
in  the  last  paragraph  varies  with  the  position  of  the  ob- 
server on  the  surface  of  the  earth,  and  has  its  greatest 
value  when  the  body  is  seen  in  the  horizon  of  the  ob- 
server, as  may  be  seen  by  an  inspection  of  Fig.  19,  in 
which  the  angle  GPS  attains  its  maximum  when  the  line 
P  8  is  tangent  to  the  earth's  surface,  in  which  case  P 
will  appear  in  the  horizon  of  the  observer  at  S. 


FlG.    19. — HORIZONTAL   PARALLAX. 

The  horizontal  parallax  depends  upon  the  distance  of  a 
body  in  the  following  manner  :  In  the  triangle  C  P  $, 
right-angled  at  S,  we  have 

OS=  GP&m  CPS. 

If,  then,  we  put 

p,  the  radius  of  the  earth  C  S ; 

r,  the  distance  of  the  body  P  from  the  centre  of  the 
earth  ; 

TT,  the  angle  S  P  G,  or  the  horizontal  parallax, 
we  shall  have, 

P 

p  =  r  sin  n  ;    r  =  — 

sin  n 

Since  the  earth  is  not  perfectly  spherical,  the  quantity  p 
is  not  absolutely  constant  for  all  parts  of  the  earth,  and  its 
greatest  value  is  usually  taken  as  that  to  which  the  hori- 
zontal value  shall  be  referred.  This  greatest  value  is,  as 
we  shall  hereafter  see,  the  radius  of  the  equator,  and  the 


52  ASTRONOMY. 

corresponding  value  of  the  parallax  is  therefore  called  the 
equatorial  horizontal  parallax. 

When  the  distance  r  of  the  body  is  known,  the  equa- 
torial horizontal  parallax  can  be  found  by  the  first  of  the 
above  equations  ;  when  the  parallax  can  be  observed,  the 
distance  r  is  found  from  the  second  equation.  How  this 
is  done  will  be  described  in  treating  the  subject  of  celes- 
tial measurement. 

It  is  easily  seen  that  the  equatorial  horizontal  parallax, 
or  the  angle  C  P  S,  is  the  same  as  the  angular  semi- 
diameter  of  the  earth  seen  from  the  object  P.  In  fact, 
if  we  draw  the  line  P  Sf  tangent  to  the  earth  at  S' ,  the 
angle  S  P  S'  will  be  the  apparent  angular  diameter  of  the 
earth  as  seen  from  P,  and  will  also  be  double  the  angle 
GPS.  The  apparent  semi-diameter  of  a  heavenly  body 
is  therefore  given  by  the  same  formulae  as  the  parallax, 
its  own  radius  being  substituted  for  that  of  the  earth.  If 
we  put, 

/o,  the  radius  of  the  body  in  linear  measure  ; 

r,  the  distance  of  its  centre  from  the  observer,  expressed 
in  the  same  measure  ; 

s,  its  angular  semi-diameter,  as  seen  by  the  observer  ; 
we  shall  have, 

sin  s  —  — . 
r 

If  we  measure  the  semi-diameter  s,   and  know  the  dis- 
tance, /•,  the  radius  of  the  body  will  be 

p  =  r  sin  s. 

Generally  the  angular  semi-diameters  of  the  heavenly 
bodies  are  so  small  that  they  may  be  considered  the  same 
as  their  sines.  We  may  therefore  say  that  the  apparent 
angular  diameter  of  a  heavenly  body  varies  inversely  as 
its  distance. 


CHAPTER  II. 

ASTRONOMICAL  INSTRUMENTS. 
§  1.    THE   REFRACTING  TELESCOPE. 

IN  explaining  the  theory  and  use  of  the  refracting  tele- 
scope, we  shall  assume  that  the  reader  is  acquainted  with 
the  fundamental  principles  of  the  refraction  and  disper- 
sion of  light,  so  that  the  simple  enumeration  of  them 
will  recall  them  to  his  mind.  These  principles,  so  far 
as  we  have  occasion  to  refer  to  them,  are,  that  when 
a  ray  of  light  passing  through  a  vacuum  enters  a  trans- 
parent medium,  it  is  refracted  or  bent  from  its  course 
in  a  direction  toward  a  line  perpendicular  to  the  sur- 
face at  the  point  where  the  ray  enters  ;  that  this  bend- 
ing follows  a  certain  law  known  as  the  law  of  sines  ; 
that  when  a  pencil  of  rays  emanating  from  a  luminous 
point  falls  nearly  perpendicularly  upon  a  convex  lens, 
the  rays,  after  passing  through  it,  all  converge  toward  a 
point  on  the  other  side  called  a  focus  ;  that  light  is  com- 
pounded of  rays  of  various  degrees  of  refrangibility,  BO 
that,  when  thus  refracted,  the  component  rays  pursue 
slightly  different  courses,  and  in  passing  through  a  lens 
come  to  slightly  different  foci  ;  and  finally,  that  the  ap- 
parent angular  magnitude  subtended  by  an  object  when 
viewed  from  any  point  is  inversely  proportional  to  its 
distance.* 

*  More  exactly,  in  the  case  of  a  globe,  the  sine  of  the  angle  is  in- 
versely as  the  distance  of  the  object,  as  shown  on  the  preceding  page. 


ASTRONOMY. 

We   shall   first  describe  "the  telescope  in   its  simplest 
form,  showing  the  principles  upon  which 
its  action  depends,  leaving  out  of  considera- 
tion the  defects  of  aberration  which  require 
special  devices  in  order  to  avoid  them.     In 
the  simplest  form  in  which  we  can  conceive 
g  of  a  telescope,  it  consists  of  two  lenses  of 
3  unequal  focal  lengths.     The  purpose  of  one 
°  of  these  lenses  (called  the  objective,  or  object 
§  glass)  is  to  bring  the  rays  of  light  from  a 
22  distant  object   at   which   the   telescope   is 
^  pointed,  to  a  focus  and  there   to  form  an 
§  image  of  the  object.     The  purpose  of  the 
w  other  lens  (called  the  eye-piece)  is  to  view 
g  this  object,  or,  more  precisely,  to  form  an- 
£  other  enlarged  image  of  it  on  the  retina  of 

*  the  eye. 

g  The  figure  gives  a  representation  of  the 
«  course  of  one  pencil  of  the  rays  which  go  to 
fe  form  the  image  A  1'  of  an  object  /  B  after 

*  passing  through  the  objective  0  0 '.     The 
>  pencil  chosen  is  that  composed  of  all  the 
g  rays  emanating  from  I  which  can  possibly 
B  fall  on  the  objective  0  0'.     All  these  are, 
fc  by  the  action  of  the  objective,  concentrated 
°  at  the  point  I'.    In  the  same  way  each  point 

2  of  the  image  out  of  the  optical  axis  A  B 

3  emits  an  oblique  pencil  of  diverging  rays 
J:  which  are  made  to  converge  to  some  point 
°!  of  the  image  by  the  lens.     The  image  of 

|  £  the  point  B  of  the  object  is  the  point  A  of 
the  image.  We  must  conceive  the  image  of 
any  object  in  the  focus  of  any  lens  (or 
mirror)  to  be  formed  by  separate  bundles 
of  rays  as  in  the  figure.  The  image  thus 
formed  becomes,  in  its  turn,  an  object  to 
be  viewed  by  the  eye-piece.  After  the  rays  meet  to  form 


MAGNIFYING  POWER  OF  TELESCOPE.  55 

the  image  of  an  object,  as  at  T7,  they  continue  on  their 
course,  diverging  from  T  as  if  the  latter  were  a  material 
object  reflecting  the  light.  There  is,  however,  this  excep- 
tion :  that  the  rays,  instead  of  diverging  in  every  direction, 
only  form  a  small  cone  having  its  vertex  at  7',  and  having 
its  angle  equal  to  01'  0'.  The  reason  of  this  is  that 
only  those  rays  which  pass  through  the  objective  can  form 
the  image,  and  they  must  continue  on  their  course  in 
straight  lines  after  forming  the  image.  This  image  can 
now  be  viewed  by  a  lens,  or  even  by  the  unassisted  eye,  if 
the  observer  places  himself  behind  it  in  the  direction  A, 
so  that  the  pencil  of  rays  shall  enter  his  eye.  For  the  pres- 
ent we  may  consider  the  eye-piece  as  a  simple  lens  of 
short  focus  like  a  common  hand-magnifier,  a  more  com- 
plete description  being  given  later. 

Magnifying  Power.— To  understand  the  manner  in 
which  the  telescope  magnifies,  we  remark  that  if  an  eye  at 
the  object-glass  could  view  the  image,  it  would  appear  of 
the  same  size  as  the  actual  object,  the  image  and  the  object 
subtending  the  same  angle,  but  lying  in  opposite  direc- 
tions. This  angular  magnitude  being  the  same,  whatever 
the  focal  distance  at  which  the  image  is  formed,  it  follows 
that  the  size  of  the  image  varies  directly  as  the  focal  length 
of  the  object-glass.  But  when  we  view  an  object  with  a 
lens  of  small  focal  distance,  its  apparent  magnitude  is  the 
same  as  if  it  were  seen  at  that  focal  distance.  Consequently 
the  apparent  angular  magnitude  will  be  inversely  as  the 
focal  distance  of  the  lens.  Hence  the  focal  image  as 
seen  with  the  eye-piece  will  appear  larger  than  it  would 
when  viewed  from  the  objective,  in  the  ratio  of  the  focal 
distance  of  the  objective  to  that  of  the  eye-piece.  But  we 
have  said  that,  seen  through  the  objective,  the  image  and 
the  real  object  subtend  the  same  angle.  Hence  the  angu- 
lar magnifying  power  is  equal  to  the  focal  distance  of  the 
objective,  divided  by  that  of  the  eye-piece.  If  we  simply 
turn  the  telescope  end  for  end,  the  objective  becomes  the 
eye-piece  and  the  latter  the  objective.  The  ratio  is  in- 


5G  ASTRONOMY. 

verted,  and  the  object  is  diminished  in  size  in  the  same 
ratio  that  it  is  increased  when  viewed  in  the  ordinary 
way.  If  we  should  form  a  telescope  of  two  lenses  of 
equal  focal  length,  by  placing  them  at  double  their  focal 
distance,  it  would  not  magnify  at  all. 

The  image  formed  by  a  convex  lens,  being  upside 
down,  and  appearing  in  the  same  position  when  viewed 
with  the  eye-piece,  it  follows  that  the  telescope,  when 
constructed  in  the  simplest  manner,  shows  all  objects  in- 
verted, or  upside  down,  and  right  side  left.  This  is  the 
case  with  all  refracting  telescopes  made  for  astronomical 
uses. 

Light-gathering  Power.— It  is  not  merely  by  magnify- 
ing that  the  telescope  assists  the  vision,  but  also  by  in- 
creasing the  quantity  of  light  which  reaches  the  eye  from 
the  object  at  which  we  look.  Indeed,  should  we  view  an 
object  through  an  instrument  which  magnified,  but  did 
not  increase  the  amount  of  light  received  by  the  eye,  it  is 
evident  that  the  brilliancy  would  be  diminished  in  propor- 
tion as  the  surface  of  the  object  was  enlarged,  since  a  con- 
stant amount  of  light  would  be  spread  over  an  increased 
surface  ;  and  thus,  unless  the  light  were  faint,  the  object 
might  become  so  darkened  as  to  be  less  plainly  seen  than 
with  the  naked  eye.  How  the  telescope  increases  the 
quantity  of  light  will  be  seen  by  considering  that  when  the 
unaided  eye  looks  at  any  object,  the  retina  can  only  re- 
ceive so  many  rays  as  fall  upon  the  pupil  of  the  eye.  By 
the  use  of  the  telescope,  it  is  evident  that  as  many  rays 
can  be  brought  to  the  retina  as  fall  on  the  entire  object- 
glass.  The  pupil  of  the  human  eye,  in  its  normal  state, 
has  a  diameter  of  about  one  fifth  of  an  inch  ;  and  by  the 
use  of  the  telescope  it  is  virtually  increased  in  surface  in 
the  ratio  of  the  square  of  the  diameter  of  the  objective  to 
the  square  of  one  fifth  of  an  inch.  Thus,  with  a  two-inch 
aperture  to  our  telescope,  the  number  of  rays  collected  is 
one  hundred  times  as  great  as  the  number  collected  with 
the  naked  eye. 


POWER  OF  TELESCOPE.  57 

With  a  5-inch  object-glass,  the  ratio  is  625  to  1 

"      10    "         "          "         "       "  2,500  to  1 

"      15    "         "          "         "       "  5,625  to  1 

"     20    "        "          "         "       "  10,000  to  1 

"     26    "        "          "         "       "  16,900  to  1 

When  a  minute  object,  like  a  star,  is  viewed,  it  is 
necessary  that  a  certain  number  of  rays  should  fall  on  the 
retina  in  order  that  the  star  may  be  visible  at  all.  It  is 
therefore  plain  that  the  use  of  the  telescope  enables  an 
observer  to  see  much  fainter  stars  than  he  could  detect 
with  the  naked  eye,  and  also  to  see  faint  objects  much 
better  than  by  unaided  vision  alone.  Thus,  with  a  26- 
inch  telescope  we  may  see  stars  so  minute  that  it  would 
require  many  thousands  to  be  visible  to  the  unaided  eye. 

An  important  remark  is,  however,  to  be  made  here. 
Inspecting  Fig.  20  we  see  that  the  cone  of  rays  passing 
through  the  object-glass  converges  to  a  focus,  then  diverges 
at  the  same  angle  in  order  to  pass  through  the  eye-piece. 
After  this  passage  the  rays  emerge  from  the  eye-piece 
parallel,  as  shown  in  Fig.  22.  It  is  evident  that  the 
diameter  of  this  cylinder  of  parallel  rays,  or  "  emergent 
pencil,"  as  it  is  called,  is  less  than  the  diameter  of  the 
object-glass,  in  the  same  ratio  that  the  focal  length  of  the 
eye-piece  is  less  than  that  of  the  object-glass.  For  the 
central  ray  1 1'  is  the  common  axis  of  two  cones,  A  1'  and 
0  I'  Ofj  having  the  same  angle,  and  equal  in  length  to 
the  respective  focal  distances  of  the  glasses.  But  this 
ratio  is  also  the  magnifying  power.  Hence  the  diameter 
of  the  emergent  pencil  of  rays  is  found  by  dividing  the 
diameter  of  the  object-glass  by  the  magnifying  power. 
Now  it  is  clear  that  if  the  magnifying  power  is  so  small 
that  this  emergent  pencil  is  larger  than  the  pupil  of  the 
eye,  all  the  light  which  falls  on  the  object-glass  cannot 
enter  the  pupil.  This  will  be  the  case  whenever  the 
magnifying  power  is  less  than  five  for  every  inch  of 
aperture  of  the  glass.  If,  for  example,  the  observer  should 


58  ASTRONOMY. 

look  through  a  twelve-inch  telescope  with  an  eye-piece 
so  large  that  the  magnifying  power  was  only  30,  the 
emergent  pencil  would  be  two  fifths  of  an  inch  in  diam- 
eter, and  only  so  much  of  the  light  could  enter  the  pupil 
as  fell  on  the  central  six  inches  of  the  object-glass. 
Practically,  therefore,  the  observer  would  only  be  using  a 
six-inch  telescope,  all  the  light  which  fell  outside  of  the 
six-inch  circle  being  lost.  In  order,  therefore,  that  he 
may  get  the  advantage  of  all  his  object-glass,  he  must  use 
a  magnifying  power  at  least  five  times  the  diameter  of  his 
objective  in  inches. 

When  the  magnifying  power  is  carried  beyond  this 
limit,  the  action  of  a  telescope  will  depend  partly  on  the 
nature  of  the  object  one  is  looking  at.  Viewing  a  star, 
the  increase  of  power  will  give  no  increase  of  light,  and 
therefore  no  increase  in  the  apparent  brightness  of  the 
star.  If  one  is  looking  at  an  object  having  a  sensible 
surface,  as  the  moon,  or  a  planet,  the  light  coming 
from  a  given  portion  of  the  surface  will  be  spread  over  a 
larger  portion  of  the  retina,  as  the  magnifying  power 
is  increased.  All  magnifying  must  then  be  gained  at 
the  expense  of  the  apparent  illumination  of  the  surface. 
Whether  this  loss  of  illumination  is  important  or  not  will 
depend  entirely  on  how  much  light  is  to  spare.  In  a 
general  way  we  may  say  that  the  moon  and  all  the  plan- 
ets nearer  than  Saturn  are  so  brilliantly  illuminated  by 
the  sun  that  the  magnifying  power  can  be  carried  many 
times  above  the  limit  without  any  loss  in  the  distinctness 
of  vision. 

The  Telescope  in  Measurement. — A  telescope  is  gen- 
erally thought  of  only  as  an  instrument  to  assist  the  eye 
by  its  magnifying  and  light-gathering  power  in  the  man- 
ner we  have  described.  But  it  has  a  very  important 
additional  function  in  astronomical  measurements  by  en- 
abling the  astronomer  to  point  at  a  celestial  object  with  a 
certainty  and  accuracy  otherwise  unattainable.  This  func- 
tion of  the  telescope  was  not  recognized  for  more  than 


USE  OF  TELESCOPE.  59 

half  a  century  after  its  invention,  and  after  a  long  and 
rather  acrimonious  contest  between  two  schools  of  astron- 
omers. Until  the  middle  of  the  seventeenth  century, 
when  an  astronomer  wished  to  determine  the  altitude  of  a 
celestial  object,  or  to  measure  the  angular  distance  be- 
tween two  stars,  he  was  obliged  to  point  his  quadrant  or 
other  measuring  instrument  at  the  object  by  means  of 
"  pinnules. "  These  served  the  same  purpose  as  the  sights 
on  a  rifle.  In  using  them,  however,  a  difficulty  arose. 
It  was  impossible  for  the  observer  to  have  distinct  vision 
both  of  the  object  and  of  the  pinnules  at  the  same  time, 
because  when  the  eye  was  focused  on  either  pinnule,  or 
on  the  object,  it  was  necessarily  out  of  focus  for  the 
others.  The  only  way  to  diminish  this  difficulty  was  to 
lengthen  the  arm  on  which  the  pinnules  were  fastened  so 
that  the  latter  should  be  as  far  apart  as  possible.  Thus 
TYCHO  BRAKE,  before  the  year  1600,  had  measuring  in- 
struments very  much  larger  than  any  in  use  at  the  pres- 
ent time.  But  this  plan  only  diminished  the  difficulty  and 
could  not  entirely  obviate  it,  because  to  be  manageable 
the  instrument  must  not  be  very  large. 

About  1670  the  English  and  French  astronomers  found 
that  by  simply  inserting  fine  threads  or  wires  exactly  in 
the  focus  of  the  telescope,  and  then  pointing  it  at  the  ob- 
ject, the  image  of  that  object  formed  in  the  focus  could  be 
made  to  coincide  with  the  threads,  so  that  the  observer 
could  see  the  two  exactly  superimposed  upon  each  other. 
When  thus  brought  into  coincidence,  it  was  known  that 
the  point  of  the  object  on  which  the  wires  were  set  was  in 
a  straight  line  passing  through  the  wires,  and  through  the 
centre  of  the  object-glass.  So  exactly  could  such  a  point- 
ing be  made,  that  if  the  telescope  did  not  magnify  at  all 
(the  eye-piece  and  object-glass  being  of  equal  focal  length), 
a  very  important  advance  would  still  be  made  in  the  ac- 
curacy of  astronomical  measurements.  This  line,  passing 
centrally  through  the  telescope,  we  call  the  line  of  col- 
limation  of  the  telescope,  A  B  in  Fig.  20.  If  we  have 


00  ASTRONOMY. 

any  way  of  determining  it  we*  at  once  realize  the  idea  ex- 
pressed in  the  opening  chapter  of  this  book,  of  a  pencil  ex- 
tended in  a  definite  direction  from  the  earth  to  the  heav- 
ens. If  the  observer  simply  sets  his  telescope  in  a  fixed 
position,  looks  through  it  and  notices  what  stars  pass  along 
the  threads  in  the  eye-piece,  he  knows  that  those  stars  all 
lie  in  the  line  of  collimation  of  his  telescope  at  that  instant. 
By  the  diurnal  motion,  a  pencil-mark,  as  it  were,  is  thus 
being  made  in  the  heavens,  the  direction  of  which  can  be 
determined  with  far  greater  precision  than  by  any  meas- 
urements with  the  unaided  eye.  The  direction  of  this  line 
of  collimation  can  be  determined  by  methods  which  we 
need  not  now  describe  in  detail. 

The  Achromatic  Telescope. — The  simple  form  of  tele- 
scope which  we  have  described  is  rather  a  geometrical 
conception  than  an  actual  instrument.  Only  the  earli- 
est instruments  of  this  class  were  made  with  so  few  as  two 
lenses.  GALILEO'S  telescope  was  not  made  in  the  form 
which  we  have  described,  for  instead  of  two  convex  lenses 
having  a  common  focus,  the  eye-piece  was  concave,  and 
was  placed  at  the  proper  distance  inside  of  the  focus  of  the 
objective.  This  form  of  instrument  is  still  used  in  opera- 
glasses,  but  is  objectionable  in  large  instruments,  owing  to 
the  smallness  of  the  field  of  view.  The  use  of  two  con- 
vex lenses  was,  we  believe,  first  proposed  by  KEPLER. 
Although  telescopes  of  this  simple  form  were  wonderful 
instruments  in  their  day,  yet  they  would  not  now  be  re- 
garded as  serving  any  of  the  purposes  of  such  an  instru- 
ment, owing  to  the  aberrations  with  which  a  single  lens  is 
affected.  We  know  that  when  ordinary  light  passes 
through  a  simple  lens  it  is  partially  decomposed,  the  differ- 
ent rays  coming  to  a  focus  at  different  distances.  The 
focus  for  red  rays  is  most  distant  from  the  object-glass, 
and  that  for  violet  rays  the  nearest  to  it.  Thus  arises 
the  chromatic  aberration  of  a  lens.  But  this  is  not  all. 
Even  if  the  light  is  but  of  a  single  degree  of  refrangi- 
bility,  if  the  surfaces  of  our  lens  are  spherical,  the  rays 


ACHROMATIC  OBJECT-GLASS.  61 

which  pass  near  the  edge  will  come  to  a  shorter  focus 
than  those  which  pass  near  the  centre.  Thus  arises 
spherical  aberration.  This  aberration  might  be  avoided 
if  lenses  could  be  ground  with  a  proper  gradation  of 
curvature  from  the  centre  to  the  circumference.  Prac- 
tically, however,  this  is  impossible  ;  the  deviation  from 
uniform  sphericity,  which  an  optician  can  produce,  is  too 
small  to  neutralize  the  defect. 

Of  these  two  defects,  the  chromatic  aberration  is  much 
the  more  serious  ;  and  no  way  of  avoiding  it  was  known 
until  the  latter  part  of  the  last  century.  The  fact  had, 
indeed,  been  recognized  by  mathematicians  and  physicists, 
that  if  two  glasses  could  be  found  having  very  different 
ratios  of  refractive  to  dispersive  powers,*  the  defect  could 
be  cured  by  combining  lenses  made  of  these  different 
kinds  of  glass.  But  this  idea  was  not  realized  until  the 
time  of  DOLLOND,  an  English  optician  who  lived  during 
the  last  century.  This  artist  found  that  a  concave  lens  of 
flint  glass  could  be  combined  with  a  convex  lens  of  crown  of 
double  the  curvature  in  such  a  manner  that  the  dispersive 
powers  of  the  two  lenses  should  neutralize  each  other,  being 
equal  and  acting  in  opposite  di- 
rections. But  the  crown  glass 
having  the  greater  refractive 
power,  owing  to  its  greater  cur- 
vature, the  rays  would  be  brought 
to  a  focus  without  dispersion. 
Such  is  the  construction  of  the  FIG.  21.— SECTION  OF  OBJECT- 
achromatic  objective.  As  now  GLASS. 

made,  the  outer  or  crown  glass  lens  is  double  convex  ;  the 
inner  or  flint  one  is  generally  nearly  plano-concave. 
Fig.  21  shows  the  section  of  such  an  objective  as  made 
by  ALVAN  CLARK  &  SONS,  the  inner  curves  of  the  crown 
and  flint  being  nearly  equal. 

*  By  the  refractive  power  of  a  glass  is  meant  its  power  of  bending  the 
rays  out  of  their  course,  so  as  to  bring  them  to  a  focus.  By  its  disper- 
sive power  is  meant  its  power  of  separating  the  colors  so  as  to  form  a 
spectrum,  or  to  produce  chromatic  aberration. 


62  ASTRONOMY. 

A  great  advantage  of  the  a'chromatic  objective  is  that  it 
may  be  made  to  correct  the  spherical  as  well  as  the  chro- 
matic aberration.  This  is  effected  by  giving  the  proper 
curvature  to  the  various  surfaces,  and  by  making  such 
slight  deviations  from  perfect  sphericity  that  rays  passing 
through  all  parts  of  the  glass  shall  come  to  the  same  focus. 

The  Secondary  Spectrum. — It  is.  now  known  that  the 
chromatic  aberration  of  an  objective  cannot  be  perfectly 
corrected  with  any  combination  of  glasses  yet  discovered. 
In  the  best  telescopes  the  brightest  rays  of  the  spectrum, 
which  are  the  yellow  and  green  ones,  are  all  brought  to 
the  same  focus,  but  the  red  and  blue  ones  reach  a  focus 
a  little  farther  from  the  objective,  and  the  violet  ones  a 
focus  still  farther.  Hence,  if  we  look  at  a  bright  star 
through  a  large  telescope,  it  will  be  seen  surrounded  by  a 
blue  or  violet  light.  If  we  push  the  eye-piece  in  a  little 
the  enlarged  image  of  the  star  will  be  yellow  in  the  centre 
and  purple  around  the  border.  This  separation  of  colors 
by  a  pair  of  lenses  is  called  a  secondary  spectrum. 

Eye-Piece. — In  the  skeleton  form  of  telescope  before 
described  the  eye-piece  as  well  as  the  objective  was  con- 
sidered as  consisting  of  but  a  single  lens.  But  with  such 
an  eye -piece  vision  is  imperfect,  except  in  the  centre  of 
the  field,  from  the  fact  that  the  image  does  not  throw 
rays  in  every  direction,  but  only  in  straight  lines  away 
from  the  objective.  Hence,  the  rays  from  near  the  edges 
of  the  focal  image  fall  on  or  near  the  edge  of  the  eye- 
piece, whence  arises  distortion  of  the  image  formed  on 
the  retina,  and  loss  of  light.  To  remedy  this  difficulty  a 
lens  is  inserted  at  or  very  near  the  place  where  the  focal 
image  is  formed,  for  the  purpose  of  throwing  the  different 
pencils  of  rays  which  emanate  from  the  several  parts  of 
the  image  toward  the  axis  of  the  telescope,  so  that  they 
shall  all  pass  nearly  through  the  centre  of  the  eye  lens  pro- 
per. These  two  lenses  are  together  called  the  eye-piece. 

There  are  some  small  differences  of  detail  in  the  con- 
struction of  eye-pieces,  but  the  general  principle  is  the 


THEORY  OF  OBJECT-GLASS.  63 

same  in  all.  The  two  recognized  classes  are  the  posi- 
tive and  negative,  the  former  being  x  those  in  which  the 
image  is  formed  before  the  light  reaches  the  field  lens  ;  the 
negative  those  in  which  it  is  formed  between  the  lenses. 

The  figure  shows  the  positive  eye-piece  drawn  accurately  to  scale. 
0  I  is  one  of  the  converging  pencils  from  the  object-glass  which 
forms  one  point  (/)  of  the  focal  image  la.  This  image  is  viewed 
by  the  field  lens  F  of  the  eye-piece  as  a  real  object,  and  the  shaded 
pencil  between  F  and  E  shows  the  course  of  these  rays  after  de- 
viation by  F.  If  there  were  no  eye-lens  E  an  eye  properly  placed 
beyond  F  would  see  an  image  at  /'  a'.  The  eye-lens  E  receives  the 
pencil  of  rays,  and  deviates  it  to  the  observer's  eye  placed  at  such  a 
point  that  the  whole  incident  pencil  will  pass  through  the  pupil 
and  fall  on  the  retina,  and  thus  be  effective.  As  we  saw  in  the 


22. — SECTION   OP   A   POSITIVE   EYE  PTRCR. 


figure  of  the  refracting  telescope,  every  point  of  the  object  produces 
a  pencil  similar  to  0  /,  and  the  whole  surfaces  of  the  lenses  F 
and  E  are  covered  with  rays.  All  of  these  pencils  passing  through 
the  pupil  go  to  make  up  the  retinal  image.  This  image  is  referred 
by  the  mind  to  the  distance  of  distinct  vision  (about  ten  inches), 
and  the  image  A  I"  represents  the  dimension  of  the  final  image 

A   T' 
relative  to  the  image  a  I  as  formed  by  the  objective  and  — =-  is 

evidently  the  magnifying  power  of  this  particular  eye-piece  used 
in  combination  with  this  particular  objective. 

More  Exact  Theory  of  the  Objective. — For  the  benefit  of  the 
reader  who  wishes  a  more  precise  knowledge  of  the  optical  princi- 
ples on  which  the  action  of  the  objective  or  other  system  of  lenses 
depends,  we  present  the  following  geometrical  theory  of  the  sub- 
ject. This  theory  is  not  rigidly  exact,  but  is  sufficiently  so  for  all 
ordinary  computations  of  the  focal  lengths  and  sizes  of  image  in 
the  usual  combinations  of  lenses. 


64  ASTRONOMY. 

Centres  of  Convergence  and  Divergence. — Suppose  A  B,  Fig. 
23,  to  be  a  lens  or  combination  of  lenses  on  which  the  light  falls  from 
the  left  hand  and  passes  through  to  the  right.  Suppose  rays  parallel 
to  7?  P  to  fall  on  every  part  of  the  first  surface  of  the  glass.  After 
passing  through  it  they  are  all  supposed  to  converge  nearly  or  ex- 
actly to  the  same  point  R' .  Among  all  these  rays  there  is  one,  and 
one  only,  the  course  of  which,  after  emerging  from  the  glass  at  Q, 
will  be  parallel  to  its  original  direction  RP.  Let  R  P  Q  R'  be  this 
central  ray,  which  will  be  completely  determined  by  the  direction 
from  which  it  comes.  Next,  let  us  take  a  ray  coming  from  another 
direction  as  8  P.  Among  all  the  rays  parallel  to  8  P,  let  us  take 
that  one  which,  after  emerging  from  the  glass  at  T,  moves  in  a  line 
parallel  to  its  original  direction.  Continuing  the  process,  let  us 
suppose  isolated  rays  coming  from  all  parts  of  a  distant  object  sub- 
ject to  the  single  condition  that  the  course  of  each,  after  passing 
through  the  glass  or  system  of  glasses,  shall  be  parallel  to  its  original 
course.  These  rays  we  may  call  central  rays.  They  have  this  re- 
markable property,  pointed  out  by  GAUSS  :  that  they  all  converge 


FIG.  23. 

toward  a  single  point,  P,  in  coming  to  the  glass,  and  diverge  from 
another  point,  P,  after  passing  through  the  last  lens.  These  points 
were  termed  by  GAUSS  u  Hauptpunkte, "  or  principal  points.  But 
they  will  probably  be  better  understood  if  we  call  the  first  one  the 
centre  of  convergence,  and  the  second  the  centre  of  divergence. 
It  must  not  be  understood  that  the  central  rays  necessarily  pass 
through  these  centres.  If  one  of  them  lies  outside  the  first  or  last 
refracting  surface,  then  the  central  rays  must  actually  pass  through 
it.  But  if  they  lie  between  the  surfaces,  they  will  be  fixed  by  the 
continuation  of  the  straight  line  in  which  the  rays  move,  the  latter 
being  refracted  out  of  their  course  by  passing  through  the  surface, 
and  thus  avoiding  the  points  in  question.  If  the  lens  or  system  of 
lenses  be  turned  around,  or  if  the  light  passes  through  them  in  an 
opposite  direction,  the  centre  of  convergence  in  the  first  case  be- 
comes the  centre  of  divergence  in  the  second,  and  vice  versa.  The 
necessity  of  this  will  be  clearly  seen  by  reflecting  that  a  return  ray 
of  light  will  always  keep  on  the  course  of  the  original  ray  in  the 
opposite  direction. 


THEORY  OF  OBJECT-GLASS.  65 

The  figure  represents  a  plano-convex  lens  with  light  falling  on 
the  convex  side.  In  this  case  the  centre  of  convergence  will  be 
on  the  convex  surface,  and  that  of  divergence  inside  the  glass 
about  one  third  or  two  fifths  of  the  way  from  the  convex  to  the 
plane  surface,  the  positions  varying  with  the  refractive  index  of  the 
glass.  In  a  double  convex  lens,  both  points  will  lie  inside  the  glass, 
while  if  a  glass  is  concave  on  one  side  and  convex  on  the  other, 
one  of  the  points  will  be  outside  the  glass  on  the  eoncave  side.  It 
must  be  remembered  that  the  positions  of  these  centres  of  conver- 
gence and  divergence  depend  solely  on  the  form  and  size  of  the 
lenses  and  their  refractive  indices,  and  do  not  refer  in  any  way  to 
the  distances  of  the  objects  whose  images  they  form. 

The  principal  properties  of  a  lens  or  objective,  by  which  the  size 
of  images  are  determined,  are  as  follows  :  Since  the  angle  S'  P  R' 
made  by  the  diverging  rays  is  equal  to  R  P  8,  made  by  the  con- 
verging ones,  it  follows,  that  if  a  lens  form  the  image  of  an  object, 
the  size  of  the  image  will  be  to  that  of  the  object  as  their  respec- 
tive distances  from  the  centres  of  convergence  and  divergence.  In 
other  words,  the  object  seen  from  the  centre  of  convergence  P  will 
be  of  the  same  angular  magnitude  as  the  image  seen  from  the 
centre  of  divergence  P*. 

By  conjugate  foci  of  a  lens  or  system  of  lenses  we  mean  a  pair  of 
points  such  that  if  rays  diverge  from  the  one,  they  will  converge  to 
the  other.  Hence  if  an  object  is  in  one  of  a  pair  of  such  foci,  the 
image  will  be  formed  in  the  other. 

By  the  refractive  power  of  a  lens  or  combination  of  lenses,  we 
mean  its  influence  in  refracting  parallel  rays  to  a  focus  which  we 
may  measure  by  the  reciprocal  of  its  focal  distance  or  1  -±-f.  Thus, 
the  power  of  a  piece  of  plain  glass  is  0,  because  it  cannot  bring 
rays  to  a  focus  at  all.  The  power  of  a  convex  lens  is  positive,  while 
that  of  a  concave  lens  is  negative.  In  the  latter  case,  it  will  be 
remembered  by  the  student  of  optics  that  the  virtual  focus  is  on 
the  same  side  of  the  lens  from  which  the  rays  proceed.  It  is  to 
be  noted  that  when  we  speak  of  the  focal  distance  of  a  lens,  we 
mean  the  distance  from  the  centre  of  divergence  to  the  focus  for 
parallel  rays.  In  astronomical  language  this  focus  is  called  the 
stellar  focus,  being  that  for  celestial  objects,  all  of  which  we  may 
regard  as  infinitely  distant.  If,  now,  we  put 

p,  the  power  of  the  lens  ; 
/,  its  stellar  focal  distance  ; 

J*t  the  distance  of  an  object  from  the  centre  of  convergence  ; 
/",  the  distance  of  its  image  from  the  centre  of  divergence  ;  then 
the  equation  which  determines /will  be 

i      1  _L 
f+f'~f  ~P'' 
or, 

,        //'    .,    f>=    ff_ 
J      f  +  f"J        f-f 

From  these  equations  may  be  found  the  focal  length,  having  the 
distance  at  which  the  image  of  an  object  is  formed,  or  vice  versa. 


66  ASTRONOMY. 

§  2.    REFLECTING    TELESCOPES. 

As  we  have  seen,  the  most  essential  part  of  a  refracting 
telescope  is  the  objective,  which  brings  all  the  incident 
rays  from  an  object  to  one  focus,  forming  there  an  image 
of  thai  object.  In  reflecting  telescopes  (reflectors)  the 
objective  is  a  mirror  of  speculum  metal  or  silvered  glass 
ground  to.  the  shape  of  a  paraboloid.  The  figure  shows 
the  action  of  such  a  mirroi  on  a  bundle  of  parallel  rays, 
which,  after  impinging  on  it,  are  brought  by  reflection  to 
one  focus  F.  The  image  formed  at  this  focus  may  be 
viewed  with  an  eye-piece,  as  in  the  case  of  the  refracting 
telescope. 

The  eye- pieces  used  with  such  a  mirror  are  of  the  kinds 
already  described.  In  the  figure  the  eye -piece  would 


FlG.    24. — CONCAVE  MIRROR  FORMING  AN  IMAGE. 

have  to  be  placed  to  the  right  of  the  point  F^  and  the 
observer's  head  would  thus  interfere  with  the  incident 
light.  Various  devices  have  been  proposed  to  remedy  this 
inconvenience,  of  which  we  will  describe  the  two  most 
common. 

The  Newtonian  Telescope. — In  this  form  the  rays  of 
light  reflected  from  the  mirror  are  made  to  fall  on  a  small 
plane  mirror  placed  diagonally  just  before  they  reach  the 
principal  focus.  The  rays  are  thus  reflected  out  laterally 
through  an  opening  in  the  telescope  tube,  and  are  there 
brought  to  a  focus,  and  the  image  formed  at  the  point 
marked  by  a  heavy  white  line  in  Fig.  25,  instead  of  at 
the  point  inside  the  telescope  marked  by  a  dotted  line. 


REFLECTING   TELESCOPES. 


67 


This  focal  image  is  then  examined  by  means  of  an  or- 
dinary eye-piece,  the  head  of  the  observer  being  outside 
of  the  telescope  tube. 

This  device  is  the  invention  of  Sir  ISAAC  NEWTON. 


FIG.  25. 
NEWTONIAN  TELESCOPE. 


FIG.  26. 
CASSEGRAINIAN  TELESCOPE. 


The  Cassegrainian  Telescope. — In  this  form  a  second- 
ary convex  mirror  is  placed  in  the  tube  of  the  telescope 


08  ASTRONOMY. 

about  three  fourths  of  the  way  from  the  large  speculum 
to  the  focus.  The  rajs,  after  being  reflected  from  the 
large  speculum,  fall  on  this  mirror  before  reaching  the 
focus,  and  are  reflected  back  again  to  the  speculum  ;  an 
opening  is  made  in  the  centre  of  the  latter  to  let  the  rays 
pass  through.  The  position  and  curvature  of  the  secondary 
mirror  are  adjusted  so  that  the  focus  shall  be  formed  just 
after  passing  through  the  opening  in  the  speculum. 

In  this  telescope  the  observer  stands  behind  or  under 
the  speculum,  and,  with  the  eye-piece,  looks  through  the 
opening  in  the  centre,  in  the  direction  of  the  object. 
This  form  of  reflector  is  much  more  convenient  in  use 
than  the  Newtonian,  in  using  which  the  observer  has  to 
be  near  the  top  of  the  tube. 

This  form  was  devised  by  CASSEGRAIN  in  1672. 

The  advantages  of  reflectors  are  found  in  their  cheap- 
ness, and  in  the  fact  that,  supposing  the  mirrors  perfect  in 
figure,  all  the  rays  of  the  spectrum  are  brought  to  one 
focus.  Thus  the  reflector  is  suitable  for  spectroscopic  or 
photographic  researches  without  any  change  from  its  or- 
dinary form.  This  is  not  true  of  the  refractor,  since  the 
rays  by  which  we  now  photograph  (the  blue  and  violet 
rays)  are,  in  that  instrument,  owing  to  the  secondary 
spectrum,  brought  to  a  focus  slightly  different  from  that 
of  the  yellow  and  adjacent  rays  by  means  of  which  we 
see. 

Reflectors  have  been  made  as  large  as  six  feet  in  aper- 
ture, the  greatest  being  that  of  Lord  KOSSE,  but  those 
which  have  been  most  successful  have  hardly  ever  been 
larger  than  two  or  three  feet.  The  smallest  satellite  of 
Saturn  (Mimas)  was  discovered  by  Sir  WILLIAM  HERSCHEL 
with  a  four-foot  speculum,  but  all  the  other  satellites  dis- 
covered by  him  were  seen  with  mirrors  of  about  eighteen 
inches  in  aperture.  With  these  the  vast  majority  of  his 
faint  nebulae  were  also  discovered. 

The  satellites  of  Neptune  and  Uranus  were  discovered 
by  LASSELL  with  a  two -foot  speculum,  and  much  of  the 


REFLECTING   TELESCOPES.  69 

work  of  Lord  ROSSE  has  been  done  with  his  three-foot 
mirror,  instead  of  his  celebrated  six-foot  one. 

From  the  time  of  NEWTON  till  quite  recently  it  was 
usual  to  make  the  large  mirror  or  objective  out  of  specu- 
lum metal,  a  brilliant  alloy  liable  to  tarnish.  "When  the 
mirror  was  once  tarnished  through  exposure  to  the 
weather,  it  could  be  renewed  only  by  a  process  of  polish- 
ing almost  equivalent  to  figuring  and  polishing  the  mirror 
anew.  Consequently,  in  such  a  speculum,  after  the  cor- 
rect form  and  polish  were  attained,  there  was  great  diffi- 
culty in  preserving  them.  In  recent  years  this  difficulty 
has  been  largely  overcome  in  two  ways  :  first,  by  im- 
provements in  the  composition  of  the  alloy,  by  which  its 
liability  to  tarnish  under  exposure  is  greatly  diminished, 
and,  secondly,  by  a  plan  proposed  by  FOUCAULT,  which 
consists  in  making,  once  for  all,  a  mirror  of  glass  which 
will  always  retain  its  good  figure,  and  depositing  upon  it  a 
thin  film  of  silver  which,  may  be  removed  and  restored 
with  little  labor  as  often  as  it  becomes  tarnished. 

In  this  way,  one  important  defect  in  the  reflector  has 
been  avoided.  Another  great  defect  has  been  less  success- 
fully treated.  It  is  not  a  process  of  exceeding  difficulty 
to  give  to  the  reflecting  surface  of  either  metal  or  glass 
the  correct  parabolic  shape  by  which  the  incident  rays  are 
brought  accurately  to  one  focus.  But  to  maintain  this 
shape  constantly  when  the  mirror  is  mounted  in  a  tube, 
and  when  this  tube  is  directed  in  succession  to  various 
parts  of  the  sky,  is  a  mechanical  problem  of  extreme  diffi- 
culty. However  the  mirror  may  be  supported,  all  the 
unsupported  points  tend  by  their  weight  to  sag  away  from 
the  proper  position.  When  the  mirror  is  pointed  near 
the  horizon,  this  effect  of  flexure  is  quite  different  from 
what  it  is  when  pointed  near  the  zenith. 

As  long  as  the  mirror  is  small  (not  greater  than  eight  to 
twelve  inches  in  diameter),  it  is  found  easy  to  support  it 
so  that  these  variations  in  the  strains  of  flexure  have  little 
practical  effect.  As  we  increase  its  diameter  up  to  48  or 


70  ASTRONOMY. 

72  inches,  the  effect  of  flexure  rapidly  increases,  and 
special  devices  have  to  be  used  to  counterbalance  the 
injury  done  to  the  shape  of  the  mirror. 

§  3.    CHRONOMETERS   AND   CLOCKS. 

In  Chapter  L,  §  5,  we  described  how  the  right  ascen- 
sions of  the  heavenly  bodies  are  measured  by  the  times 
of  their  transits  over  the  meridian,  this  quantity  increas- 
ing by  a  minute  of  arc  in  four  seconds  of  time.  In  order 
to  determine  it  with  all  required  accuracy,  it  is  necessary 
that  the  time-pieces  with  which  it  is  measured  shall  go 
with  the  greatest  possible  precision.  There  is  no  great 
difficulty  in  making  astronomical  measures  to  a  second 
of  arc,  and  a  star,  by  its  diurnal  motion,  passes  over  this 
space  in  one  fifteenth  of  a  second  of  time.  It  is  there- 
fore desirable  that  the  astronomical  clock  shall  not  vary 
from  a  uniform  rate  more  than  a  few  hundredths  of  a 
second  in  the  course  of  a  day.  It  is  not,  however, 
necessary  that  it  should  be  perfectly  correct  ;  it  may  go 
too  fast  or  too  slow  without  detracting  from  its  char- 
acter for  accuracy,  if  the  intervals  of  time  which  it 
tells  off — hours,  minutes,  or  seconds — are  always  of  ex- 
actly the  same  length,  or,  in  other  words,  if  it  gains  or 
loses  exactly  the  same  amount  every  hour  and  every  day. 

The  time-pieces  used  in  astronomical  observation  are 
the  chronometer  and  the  clock. 

The  chronometer  is  merely  a  very  perfect  time-piece 
with  a  balance-wheel  so  constructed  that  changes  of  tem- 
perature have  the  least  possible  effect  upon  the  time  of  its 
oscillation.  Such  a  balance  is  called  a  compensation  bal- 
ance. 

The  ordinary  house  clock  goes  faster  in  cold  than  in 
warm  weather,  because  the  pendulum  rod  shortens  under 
the  influence  of  cold.  This  effect  is  such  that  the  clock 
will  gain  about  one  second  a  day  for  every  fall  of  3°  Cent. 
(5°. 4  Fahr.)  in  the  temperature,  supposing  the  pendulum 


THE  ASTRONOMICAL  CLOCK. 


71 


rod  to  be  of  iron.  Such  changes  of  rate  would  be  entirely 
inadmissible  in  a  clock  used  for  astronomical  purposes. 
The  astronomical  clock  is  therefore  provided  with  a  com- 
pensation pendulum,  by  which  the  disturbing  effects  of 
changes  of  temperature  are  avoided. 

There  are  two  forms  now  in  use,  the  Harrison  (grid- 
iron) and  the  mercurial.  In  the  gridiron  pendulum  the 
rod  is  composed  in  part  of  a  number 
of  parallel  bars  of  steel  and  brass, 
so  connected  together  that  while  the 
expansion  of  the  steel  bars  produced 
by  an  increase  of  temperature  tends 
to  depress  the  bob  of  the  pendulum, 
the  greater  expansion  of  the  brass  bars 
tends  to  raise  it.  When  the  total 
lengths  of  the  steel  and  brass  bars 
have  been  properly  adjusted  a  nearly 
perfect  compensation  occurs,  and  the 
centre  of  oscillation  remains  at  a  con- 
stant distance  from  the  point  of  sus- 
pension. The  rate  of  the  clock,  so 
far  as  it  depends  on  the  length  of  the 
pendulum,  will  therefore  be  constant. 

In  the  mercurial  pendulum  the 
weight  which  forms  the  bob  is  a 
cylindric  glass  vessel  nearly  filled 
with  mercury.  With  an  increase  of  temperature  the  steel 
suspension  rod  lengthens,  thus  throwing  the  centre  of 
oscillation  away  from  the  point  of  suspension  ;  at  the 
same  time  the  expanding  mercury  rises  in  the  cylinder, 
and  tends  therefore  to  raise  the  centre  of  oscillation. 
When  the  length  of  the  rod  and  the  dimensions  of  the 
cylinder  of  mercury  are  properly  proportioned,  the  centre 
of  oscillation  is  kept  at  a  constant  distance  from  the  point 
of  suspension.  Other  methods  of  making  this  compensa- 
tion have  been  used,  but  these  are  the  two  in  most  gen- 
eral use  for  astronomical  clocks. 


FIG.  27. — GRIDIRON 
PENDULUM. 


72  ASTRONOMY. 

The  correction  of  a  chronometer  (or  clock)  is  the  quantity  of  time 
(expressed  in  hours,  minutes,  seconds,  and  decimals  of  a  second) 
which  it  is  necessary  to  add  algebraically  to  the  indication  of  the 
hands,  in  order  that  the  sum  may  be  the  correct  time.  Thus,  if  at 
sidereal  Oh,  May  18,  at  New  York,  a  sidereal  clock  or  chronometer 
indicates  23h  58111  20s -7,  its  correction  is  +  I"1  39s- 3;  if  atO'1  (sidereal 
noon),  of  May  17,  its  correction  was  -f  I"1  38" -3,  its  daily  rate  or  the 
change  of  its  correction  in  a  sidereal  day  is  +  1s- 0:  in  other  words, 
this  clock  is  losing  I9  daily. 

For  clock  slow  the  sign  of  the  correction  is  -f- ; 
"       "     fast     4i      "     "     "  "         is  —  ; 

"      "  gaining"      '*     *'    "       rate       is  — ; 
"       "    losing    "     *•     *•    "  is  + . 

A  clock  or  chronometer  may  be  well  compensated  for  temperature, 
and  yet  its  rate  may  be  gaining  or  losing  on  the  time  it  is  intended 
to  keep :  it  is  not  even  necessary  that  the  rate  should  be  small  (ex- 
cept that  a  small  rate  is  practically  convenient),  provided  only  that 
it  is  constant.  It  is  continually  necessary  to  compute  the  clock  cor- 
rection at  a  given  time  from  its  known  correction  at  some  other  time, 
and  its  known  rate.  If  for  some  definite  instant  we  denote  the  time 
as  shown  by  the  clock  (technically  "the  clock-face")  by  T,  the  true 
time  by  T  and  the  clock  correction  by  A  T,  we  have 

T  =  T  +   A  T7,  and 
A  T  =  T'  -  T. 

In  all  observatories  and  at  sea  observations  are  made  daily  to  de- 
termine A  T.  At  the  instant  of  the  observation  the  time  T  is  noted 
by  the  clock;  from  the  data  of  the  observation  the  time  T'  is  com- 
puted. If  these  agree,  the  clock  is  correct.  If  they  differ,  A  T7  is 
found  from  the  above  equations. 

If  by  observation  we  have  found 

A  To  =  the  clock  correction  at  a  clock-time  To, 
A  T  =  the  clock  correction  at  a  clock- time  Ty 
6T  =  the  clock  rate  in  a  unit  of  time, 

we  have 

AT=  AT0  +  6T(T-  T0) 

where  T  —  T0  must  be  expressed  in  days,  hours,  etc.,  according  as 
6T  is  the  rate  in  one  day,  one  hour,  etc. 

When,  therefore,  the  clock  correction  A  To  and  rate  6T  have  been 
determined  for  a  certain  instant,  T*,  we  can  deduce  the  true  time 
from  the  clock-face  T  at  any  other  instant  by  the  equation  T'  =  T 
+  A  T0  +  6T  (T  —  To).  If  the  clock  correction  has  been  deter- 
mined at  two  different  times,  T9  and  T7  to  be  A  7 «  and  A  T,  the  rate 
is  inferred  from  the  equation 


THE  ASTRONOMICAL  CLOCK.  73 

These  equations  apply  only  so  long  as  we  can  regard  the  rate  as 
constant.  As  observations  can  be  made  only  in  clear  weather,  it  is 
plain  that  during  periods  of  overcast  sky  we  must  depend  on  these 
equations  for  our  knowledge  of  T' — /.«.,  the  true  time  at  a  clock- 
time  T. 

The  intervals  between  the  determination  of  the  clock  correction 
should  be  small,  since  even  with  the  best  clocks  and  chronometers 
too  much  dependence  must  not  be  placed  upon  the  rate.  The  follow- 
ing example  from  CHAUVENET'S  Astronomy  will  illustrate  the  practi- 
cal processes : 

"  Example. — At  sidereal  noon,  May  5,  the  correction  of  a  sidereal 
clock  is— 16'"  47S-0;  at  sidereal  noon,  May  12,  it  is  —  16'"  13S-50; 
what  is  the  sidereal  time  on  May  25,  when  the  clock-face  is  II11  13"' 
128- 6,  supposing  the  rate  to  be  uniform  ? 

May  5,  correction  =  —  16m  47s. 30 
11  12,          "          =  —  16'"  13".  50 
~~7  days'  rate    =T 33S-50 
6T=  +  4s -829. 

Taking  then  as  our  starting-point  Ta  =  May  12,  Oh,  we  have  for  the 
interval  to  T=  May  25,  llh  13m  12s- 6,  T  —  T0  =  13a  llh  13"'  12s- 6 
=  13d -467.  Hence  we  have 

ATn=  -    16m139-50 
tT  (T—  T,)  =  +      lm    5s- 03 

i  A  T  =    -    15m    8s-47 

T=  Iih13m12s-60 


T  =  lQh58m    4M3 

But  in  this  example  the  rate  is  obtained  for  one  true  sidereal  day, 
while  the  unit  of  the  interval  13d -467  is  a  sidereal  day  as  shown  by 
the  clock.  The  proper  interval  with  which  to  compute  the  rate  in 
this  case  is  13d  ,10h  58m  4s- 13  =  13d -457,  with  which  we  find 


6T  x 

13-457 

T 
T 

= 

~ 

16m 
1  ui 

13s 

4s. 

•  50 
98 
52 

•  60 

11" 

13"' 

8s. 
12s 

10" 

58'" 

4s 

•08 

This  repetition  will  be  rendered  unnecessary  by  always  giving  the  rate 
in  a  unit  of  the  clock.  Thus,  suppose  that  on  June  3,  at  4h  11'"  12s -35 
by  the  clock,  we  have  found  the  correction  +  2IU  10s -14;  and  on 
June  4,  at  14h  17'"  49s. 82  we  have  found  the  correction  +  2IU  19s- 89  ; 
the  rate  in  one  hour  of  the  dock  will  be 


74  ASTRONOMY. 

§  4.    THE    TRANSIT    INSTRUMENT. 

The  meridian  transit  instrument,  or  briefly  the  ' '  tran- 
sit, ' '  is  used  to  observe  the  transits  of  the  heavenly  bodies, 


FlG.    28. — A  TRANSIT  INSTRUMENT. 

and  from  the  times  of  these  transits  as  read  from  the 
clock  to  determine  either  the  corrections  of  the  clock  or 
the  right  ascension  of  the  observed  body,  as  explained  in 
Chapter  I. ,  §5. 


THE  TRANSIT  INSTRUMENT.  75 

It  has  two  general  forms,  one  (Fig.  28)  for  use  in  fixed 
observatories  and  one  (Fig.  29)  for  use  in  the  field. 

It  consists  essentially  of  a  telescope  T  T  (Fig.  28) 
mounted  on  an  axis  V  V  at  right  angles  to  it. 


FlG.  29. — PORTABLE   TRANSIT   INSTRUMENT. 

The  ends  of  this  axis  terminate  in  accurately  cylindrical 
steel  pivots  which  rest  in  metallic  bearings  V  V,  in  shape 
like  the  letter  Y,  and  hence  called  the  Ys. 


76  ASTRONOMY. 

These  are  fastened  to  two  pillars  of  stone,  brick,  or 
iron.  Two  counterpoises  W  W  are  connected  with  the 
axis  as  in  the  plate,  so  as  to  take  a  large  portion  of  the 
weight  of  the  axis  and  telescope  from  the  Ys,  and  thus  to 
diminish  the  friction  upon  these  and  to  render  the  rota- 
tion about  V  V  more  easy  and  regular.  In  the  ordinary 
use  of  the  transit,  the  line  V  V  is  placed  accurately  level 
and  perpendicular  to  the  meridian,  or  in  the  east  and  wTest 
line.  To  effect  this  "  adjustment,"  there  are  two  sets  of 
adjusting  screws,  by  which  the  ends  of  V  V  in  the  Ys  may 
be  moved  either  up  and  down  or  north  and  south.  The 
plate  gives  the  form  of  transit  used  in  permanent  observa- 
tories, and  shows  the  observing  chair  (7,  the  reversing  car- 
riage R,  and  the  level  Z.  The  arms  of  the  latter  have 
Y's,  which  can  be  placed  over  the  pivots  V  V. 

The  line  of  collimation  of  the  transit  telescope  is  the 
line  drawn  through  the  centre  of  the  objective  perpendic- 
ular to  the  rotation  axis  V  V. 

The  reticle  is  a  network  of  fine  spider  lines  placed  in 
the  focus  of  the  objective. 

In  Fig.  30  the  circle  represents  the  field  of  view  of  a 
transit  as  seen  through  the  eye-piece.  The  seven  ver- 
tical lines,  I,  II,  III,  IV,  Y,  VI, 
VII,  are  seven  fine  spider  lines 
tightly  stretched  across  a  metal  plate 
or  diaphragm,  and  so  adjusted  as  to 
be  perpendicular  to  the  direction  of 
a  star's  apparent  diurnal  motion. 
This  metal  plate  can  be  moved  right 
and  left  by  five  screws.  The  hori- 
zontal wires,  guide-wires,  a  and  &, 
mark  the  centre  of  the  field.  The 
field  is  illuminated  at  night  by  a  lamp  at  the  end  of  the 
axis  which  shines  through  the  hollow  interior  of  the  lat- 
ter, and  causes  the  field  to  appear  bright.  The  wires  are 
dark  against  a  bright  ground.  The  line  of  sight  is  a  line 
joining  the  centre  of  the  objective  and  the  central  one,  IY, 
of  the  seven  vertical  wires. 


THE  TRANSIT  INSTRUMENT.  77 

The  whole  transit  is  in  adjustment  when,  first,  the  axis 
V  V  is  horizontal  ;  second,  when  it  lies  east  and  west ; 
and  third,  when  the  line  of  sight  and  the  line  of  collima- 
tioii  coincide.  When  these  conditions  are  fulfilled  the 
line  of  sight  intersects  the  celestial  sphere  in  the  meridian 
of  the  place,  and  when  T  T  is  rotated  about  V  Fthe  line 
of  sight  marks  out  the  meridian  on  the  sphere. 

In  practice  the  three  adjustments  are  not  exactly  made,  since  it  is 
impossible  to  effect  them  with  mathematical  precision.  The  errors 
of  each  of  them  are  first  made  as  small  as  is  convenient,  and  are  then 
determined  and  allowed  for. 

To  find  the  error  of  level,  we  place  on  the  pivots  a  fine  level  (shown 
in  position  in  the  figure  of  the  portable  transit),  and  determine  how 
much  higher  one  pivot  is  than  the  other  in  terms  of  the  divisions 
marked  on  the  level  tube.  Such  a  level  is  shown  in  Fig.  4  of  plate 
35,  page  86.  The  value  of  one  of  these  divisions  in  seconds  of  arc 
can  be  determined  by  knowing  the  length  I  of  the  whole  level  and 
the  number  n  of  divisions  through  which  the  bubble  will  run  when 
one  end  is  raised  one  hundredth  of  an  inch. 

If  I  is  the  length  of  the  level  in  inches  or  the  radius  of  the  circle 
in  which  either  end  of  the  level  moves  when  it  is  raised,  then  as 
the  radius  of  any  circle  is  equal  to  57° -296,  3437' -75  or  206, 264" -8, 
we  have  in  this  particular  circle  one  inch  =  206, 264" -8  -T-  l\ 
0-01  inch  =  206,264-8  -4-  100  I  =  a  certain  arc  in  seconds,  say  a". 
That  is,  n  divisions  =  a",  or  one  division  d  —  a"  ~  n. 

The  error  of  collimation  can  be  found  by  pointing  the  telescope 
at  a  distant  mark  whose  image  is  brought  to  the  middle  wire.  The 
telescope  (with  the  axis)  is  then  lifted  bodily  from  the  Ys  and  re- 
placed so  that  the  axis  V  V\s>  reversed  end  for  end.  The  telescope  is 
again  pointed  to  the  distant  mark.  If  this  is  still  on  the  middle 
thread  the  line  of  sight  and  the  line  of  collimation  coincide.  If  not, 
the  reticle  must  be  moved  bodily  west  or  east  until  these  conditions 
are  fulfilled  after  repeated  reversals. 

To  find  the  error  of  azimuth  or  the  departure  of  the  direction  of 
V  V  from  an  east  and  west  line,  we  must  observe  the  transits  of 
two  stars  of  different  declinations  6  and  d,  and  right  ascensions  a 
and  a'.  Suppose  the  clock  to  be  running  correctly — that  is,  with  no 
rate — and  the  sidereal  times  of  transit  of  the  two  stars  over  the  mid- 
dle thread  to  be  0  and  0'.  If  0  —  0'  =  a  —  a',  then  the  middle  wire 
is  in  the  meridian  and  the  azimuth  is  zero.  For  if  the  azimuth 
was  not  zero,  but  the  west  end  of  the  axis  was  too  far  south,  for 
example,  the  line  of  sight  would  fall  east  of  the  meridian  for  a 
south  star,  and  further  and  further  east  the  further  south  the  star 
was.  Hence  if  the  two  stars  have  widely  different  declinations  6 
and  <J',  then  the  star  furthest  south  would  come  proportionately 
sooner  to  the  middle  wire  than  the  other,  and  B  —  0'  would  be 
different  from  a  —  a'.  The  amount  of  this  difference  gives  a 


78  ASTRONOMY. 

means  of  deducing  the  deviation  of  A  A  from  an  east  and  west 
tine.  In  a  similar  way  the  effect  of  a  given  error  of  level  on  the 
time  of  the  transit  of  a  star  of  declination  6  is  found. 

Methods  of  Observing  with  the  Transit  Instrument.— 
We  have  so  far  assumed  that  the  time  of  a  star's  transit 
over  the  middle  thread  was  known,  or  could  be  noted. 
It  is  necessary  to  speak  more  in  detail  of  how  it  is  noted. 
When  the  telescope  is  pointed  to  any  star  the  earth's 
diurnal  motion  will  carry  the  image  of  the  star  slowly 
across  the  field  of  view  of  the  telescope  (which  is  kept 
fixed),  as  before  explained.  As  it  crosses  each  of  the 
threads,  the  time  at  which  it  is  exactly  on  the  thread  is 
noted  from  the  clock,  which  must  be  near  the  transit. 

The  mean  of  these  times  gives  the  time  at  which  this 
star  was  on  the  middle  thread,  the  threads  being  at  equal 
intervals  ;  or  on  the  "  mean  thread,"  if,  as  is  the  case  in 
practice,  they  are  at  unequal  intervals. 

If  it  were  possible  for  an  astronomer  to  note  the  exact 
instant  of  the  transit  of  a  star  over  a  thread,  it  is  plain 
that  one  thread  would  be  sufficient  ;  but,  as  all  estima- 
tions of  this  time  are,  from  the  very  nature  of  the  case, 
but  approximations,  several  threads  are  inserted  in  order 
that  the  accidental  errors  of  estimations  may  be  eliminated 
as  far  as  possible.  Five,  or  at  most  seven,  threads  are 
sufficient  for  this  purpose.  In  the 
figure  of  the  reticle  of  a  transit  instru- 
ment the  star  (the  planet  Venus  in  this 
case)  may  enter  on  the  right  hand  in  the 
figure,  and  may  be  supposed  to  cross 
each  of  the  wires,  the  time  of  its  tran- 
sit over  each  of  them,  or  over  a  suffi- 
cient  number,  being  noted.  The 
method  of  noting  this  time  may  be  best 
understood  by  referring  to  the  next  figure.  Suppose  that 
the  line  in  the  middle  of  Fig.  32  is  one  of  the  transit- 
threads,  and  that  the  star  is  passing  from  the  right  hand 
of  the  figure  toward  the  left ;  if  it  is  on  this  wire  at  an 


THE  TRANSIT  INSTRUMENT.  79 

exact  second  by  the  clock  (which  is  always  near  the  ob- 
server, beating  seconds  audibly),  this  second  must  be  writ- 
ten down  as  the  time  of  the  transit  over  this  thread.  As 
a  rule,  however,  the  transit  cannot  occur  on  the  exact 
beat  of  the  clock,  but  at  the  seventeenth  second  (for  exam- 
ple) the  star  may  be  on  the  right  of  the  wire,  say  at  a  ; 
while  at  the  eighteenth  second 
it  will  have  passed  this  wire  and 
may  be  at  b.  If  the  distance  of 
a  from  the  wire  is  six  tenths  of 
the  distance  ab,  then  the  time 
of  transit  is  to  be  recorded  as  - 

hours  —  minutes    (to    be  taken 
,         , ,       ,     ,    .      ^  FIG.  32. 

from  the  clock-face),  and  seven- 
teen and  six  tenths  seconds  •  and  in  this  way  the  transit 
over  each  wire  is  observed.  This  is  the  method  of  "  eye- 
and-ear"  observation,  the  basis  of  such  work  as  we  have 
described,  and  it  is  so  called  from  the  part  which  both  the 
eye  and  the  ear  play  in  the  appreciation  of  intervals  of  time. 
The  ear  catches  the  beat  of  the  clock,  the  eye  fixes  the  place 
of  the  star  at  a  ;  at  the  next  beat  of  the  clock,  the  eye  fixes 
the  star  at  &,  and  subdivides  the  space  a  b  into  tenths,  at 
the  same  time  appreciating  the  ratio  which  the  distance 
from  the  thread  to  a  bears  to  the  distance  a  b.  This  is 
recorded  as  above.  This  method,  which  is  still  used  in 
many  observatories,  was  introduced  by  the  celebrated 
BRADLEY,  astronomer  royal  of  England  in  1750,  and  per- 
fected by  MASKELYNE,  his  successor.  A  practiced  observer 
can  note  the  time  within  a  tenth  of  a  second  in  three  cases 
out  of  four. 

There  is  yet  another  method  now  in  common  use, 
which  it  is  necessary  to  understand.  This  is  called  the 
American  or  chronographic  method,  and  consists,  in  the 
present  practice,  in  the  use  of  a  sheet  of  a  paper  wound 
about  and  fastened  to  a  horizontal  cylindrical  barrel, 
which  is  caused  to  revolve  by  machinery  once  in  one  min- 
ute of  time.  A  pen  of  glass  which  will  make  a  continu- 


80  ASTRONOMY. 

ous  line  is  allowed  to  rest  on  tlie  paper,  and  to  this  pen  a 
continuous  motion  of  translation  in  the  direction  of  the 
length  of  the  cylinder  is  given.  Now,  if  the  pen  is  allow- 
ed to  mark,  it  is  evident  that  it  will  trace  on  the  paper  an 
endless  spiral  line.  An  electric  current  is  caused  to  run 
through  the  observing  clock,  through  a  key  which  is  held 
in  the  observer's  hand  and  through  an  electro-magnet 
connected  with  the  pen. 

A  simple  device  enables  the  clock  every  second  to  give 
a  slight  lateral  motion  to  the  pen,  which  lasts  about  a 
thirtieth  of  a  second.  Thus  every  second  is  automatically 
marked  by  the  clock  on  the  chronograph  paper.  The  ob- 
server also  has  the  power  to  make  a  signal  by  his  key 
(easily  distinguished  from  the  clock-signal  by  its  different 
length),  which  is  likewise  permanently  registered  on  the 
sheet.  In  this  way,  after  the  chronograph  is  in  motion, 
the  observer  has  merely  to  notice  the  instant  at  which  the 
star  is  on  the  thread,  and  to  press  the  key  at  that  moment. 
At  any  subsequent  time,  he  must  mark  some  hour,  min- 
ute, and  second,  taken  from  the  clock,  on  the  sheet  at  its 
appropriate  place,  and  the  translation  of  the  spaces  on 
the  sheet  into  times  may  be  done  at  leisure. 

§  5.    GRADUATED   CIRCLES. 

Nearly  every  datum  in  practical  astronomy  depends 
either  directly  or  indirectly  upon  the  measure  of  an  angle. 
To  make  the  necessary  measures,  it  is  customary  to  em- 
ploy what  are  called  graduated  or  divided  circles.  These 
are  made  of  metal,  as  light  and  yet  as  rigid  as  possible, 
and  they  have  at  their  circumferences  a  narrow  flat  band 
of  silver,  gold,  or  platinum  on  which  fine  radial  lines 
called  "  divisions"  are  cut  by  a  "  dividing  engine"  at 
regular  and  equal  intervals.  These  intervals  may  be 
of  10',  5',  or  2',  according  to  the  size  of  the  circle 
and  the  degree  of  accuracy  desired.  The  narrow  band 
is  called  the  divided  limb,  and  the  circle  is  said  to  be  di- 


THE   VERNIER.  81 

vided  to  10',  5',  2'.  The  separate  divisions  are  numbered 
consecutively  from  0°  to  360°  or  from  0°  to  90°,  etc.  The 
graduated  circle  has  an  axis  at  its  centre,  and  to  this  may 
be  attached  the  telescope  by  which  to  view  the  points 
whose  angular  distance  is  to  be  determined. 

To  this  centre  is  also  attached  an  arm  which  revolves 
with  it,  and  by  its  motion  past  a  certain  number  of  divi- 
sions on  the  circle,  determines  the  angle  through  which  the 
centre  has  been  rotated.  This  arm  is  called  the  index 
arm,  and  it  usually  carries  a  vernier  on  its  extremity, 
by  means  of  which  the  spaces  on 
the  graduated  circle  are  subdivided. 
The  reading  of  the  circle  when  the 
index  arm  is  in  any  position  is  the 
number  of  degrees,  minutes,  and 
seconds  which  correspond  to  that  po- 
sition ;  when  the  index  arm  is  in  an- 
other position  there  is  a  different 
reading,  and  the  differences  of  the  two 
readings  /S"— S\  83— tf2,  #4— 8s  are  the  angles  through 
which  the  index  arm  has  turned. 

The  process  of  measuring  the  angle  between  the  objects 
by  means  of  a  divided  circle  consists  then  of  pointing  the 
telescope  at  the  first  object  and  reading  the  position  of  the 
index  arm,  and  then  turning  the  telescope  (the  index 
arm  turning  with  it)  until  it  points  at  the  second  object, 
and  again  reading  the  position  of  the  index  arm.  The 
difference  of  these  readings  is  the  angle  sought. 

To  facilitate  the  determination  of  the  exact  reading  of 
the  circle,  we  have  to  employ  special  devices,  as  the 
vernier  and  the  reading  microscope. 

The  Vernier. — In  Fig.  34,  M  N  is  a  portion  of  the 
divided  limb  of  a  graduated  circle  ;  CD  is  the  index  arm 
which  revolves  with  the  telescope  about  the  centre  of  the 
circle.  The  end  a  I  of  CD  is  also  a  part  of  a  circle  con- 
centric with  M  N)  and  it  is  divided  into  n  parts  or  divi- 
sions. The  length  of  these  n  parts  is  so  chosen  that  it  is 


82  ASTRONOMY. 

the  same  as  that  of  (n — 1)  parts  on  the  divided  limb  M  N 
or  the  reverse. 

The  first  stroke  a  is  the  zero  of  the  vernier,  and  the 
reading  is  always  determined  by  the  position  of  this  zero 
or  pointer.  If  this  has  revolved  past  exactly  twenty  di- 
visions of  the  circle,  then  the  angle  to  be  measured  is 
20  X  d,  d  being  the  value  of  one  division  on  the  limb 
(N  M)  in  arc. 


FlG.    34. — THE   VERNIER. 

Call  the  angular  value  of  one  division  on  the  vernier  d'\ 

n i  \ 

(n  —  1)  d  =  n  •  d\  or  d!  = d*  and  d— d'  =  —d> 

n  n 

d  —  d'  is  called  the  least  count  of  the  vernier  which  is  one 
7ith  part  of  a  circle  division. 

If  the  zero  a  does  not  fall  exactly  on  a  division  on  the 
circle,  but  is  at  some  other  point  (as  in  the  figure),  for  ex- 
ample between  two  divisions  whose  numbers  are  P  and 
(P  +  1),  the  whole  reading  of  the  circle  in  this  position  is 
P  x  d  +  the  fraction  of  a  division  from  P  to  a. 

If  the  mth  division  of  the  vernier  is  in  the  prolongation 
of  a  division  on  the  Iimb3  then  this  fraction  Pa  is  m 


THE  MERIDIAN  CIRCLE.  83 

(d  —  d')  —  -  •  d.      In  the  figure  n  =  10,  and  as  the  4th 

division  is  almost  exactly  in  coincidence,  m  =  4,  so  that 

4. 
the  whole  reading  of  the  circle  is  P  X  d  +  JA  •  d.    If  d  is 

10',  for  example,  and  if  the  division  P  is  numbered  297° 
40',  then  this  reading  would  be  297°  44',  the  least  count 
being  I/,  and  so  in  other  cases.  If  the  zero  had  started  from 
the  reading  280°  20',  it  must  have  moved  past  17°  24/, 
and  this  is  the  angle  which  has  been  measured. 


§  6.    THE    MERIDIAN  CIRCLE. 

The  meridian  circle  is  a  combination  of  the  transit  in- 
strument with  a  graduated  circle  fastened  to  its  axis  and 
moving  with  it.  The  meridian  circle  made  by  EEPSOLD 
for  the  United  States  Naval  Academy  at  Annapolis  is 
shown  in  the  figure.  It  has  two  circles,  c  c  and  d  c',  finely 
divided  on  their  sides.  The  graduation  of  each  circle  is 
viewed  by  four  microscopes,  two  of  which,  R  R,  are 
shown  in  the  cut.  The  microscopes  are  90°  apart.  The 
cut  shows  also  the  hanging  level  L  L,  by  which  the 
error  of  level  of  the  axis  A  A  is  found. 

The  instrument  can  be  used  as  a  transit  to  determine 
right  ascensions,  as  before  described.  It  can  be  also  used 
to  measure  declinations  in  the  following  way.  If  the  tele- 
scope is  pointed  to  the  nadir,  a  certain  division  of  the  cir- 
cles, as  JV,  is  under  the  first  microscope.  If  it  is  pointed 
to  the  pole,  the  reading  will  change  by  the  angular  distance 
between  the  nadir  and  the  pole,  or  by  90°  +  0,  0  being 
the  latitude  of  the  place  (supposed  to  be  known).  The 
polar  reading  P  is  thus  known  when  the  nadir  reading 
N  is  found.  If  the  telescope  is  then  pointed  to  various 
stars  of  unknown  polar  distances,  p' ,  p"  ^p'"y  etc.,  as  they 
successively  cross  the  meridian,  and  if  the  circle  readings 
for  these  stars  are  P ',  P" ,  P'"  ,  etc.,  it  follows  that 
p'  =  P'-P  ;  p"  =  P"  -  P  ;  p>"  =  P'"  -  P,  etc. 


84 


ASTRONOMY. 


.    35. — THE  METUTHAN   CIRCLE. 


THE  MERIDIAN  CIRCLE.  85 

To  determine  the  readings  P,  P',  P",  etc.,  we  use  the  micro- 
scopes R,  R,  etc.  The  observer,  after  having  set  the  telescope  so 
that  one  of  the  stars  shall  cross  the  field  of  view  exactly  at  its  cen- 
tre (which  may  be  here  marked  by  a  single  horizontal  thread  in 
the  reticle),  goes  to  each  of  the  microscopes  in  succession  and 
places  his  eye  at  A  (see  Fig.  1,  page  86).  He  sees  in  the  field  of  the 
microscope  the  image  of  the  divisions  of  the  graduated  scale  (Fig.  2) 
formed  at  D  (Fig.  1),  the  common  focus  of  the  lenses  A  and  C. 
Just  at  that  focus  is  placed  a  notched  scale  (Fig.  2)  and  two 
crossed  spider  lines.  These  lines  are  fixed  to  a  sliding  frame  a  a, 
which  can  be  moved  by  turning  the  graduated  head  F.  This  head 
is  divided  usually  into  sixty  parts,  each  of  which  is  1"  of  arc  on 
the  circle,  one  whole  revolution  of  the  head  serving  to  move  the 
sliding  frame  a  a,  and  its  crossed  wires  through  60''  or  V  on  the 
graduated  circle.  The  notched  scale  is  not  movable,  but  serves  to 
count  the  number  of  complete  revolutions  made  by  the  screw,  there 
being  one  notch  for  each  revolution.  The  index  i  (Fig.  2)  is  fixed, 
and  serves  to  count  the  number  of  parts  of  F  which  are  carried  past 
it  by  the  revolution  of  this  head. 

If  on  setting  the  crossed  threads  at  the  centre  of  the  motion  of 
F,  and  looking  into  the  microscope,  a  division  on  the  circle  coin- 
cides with  the  cross,  the  reading  of  the  circle  P  is  the  exact  num- 
ber of  degrees  and  minutes  corresponding  to  that  particular  divi- 
sion on  the  divided  circle. 

Usually,  however,  the  cross  has  been  apparently  carried  past  one 
of  the  exact  divisions  of  the  circle  by  a  certain  quantity,  which  is 
now  to  be  measured  and  added  to  the  reading  corresponding  to 
this  adjacent  division.  This  measure  can  be  made  by  turning  the 
screw  back  say  four  revolutions  (measured  on  the  notched  scale) 
plus  37-3  parts  (measured  by  the  index  i).  If  the  division  of  the 
circle  in  question  was  179°  50',  for  example,  the  complete  reading 
would  be  in  this  case  179°  50'  +  4'  37'' •  3  or  179°  54'  37"- 3.  Such 
a  reading  is  made  by  each  microscope,  and  the  mean  of  the  min- 
utes and  seconds  from  all  four  taken  as  the  circle  reading. 

We  now  know  how  to  obtain  the  readings  of  our  circle  when 
directed  to  any  point.  We  require  some  zero  of  reference,  as 
the  nadir  reading  (JV),  the  polar  reading  (P),  the  equator  reading, 
(§),  or  the  zenith  reading  (Z).  Any  one  of  these  being  known,  the 
circle  readings  for  any  stars  as  P,  P",  P"',  etc.,  can  be  turned  into 
polar  distances  />',  />",  p"',  etc. 

The  nadir  reading  (N)  is  the  zero  commonly  employed.  It  can 
be  determined  by  pointing  the  telescope  vertically  downward  at 
a  basin  of  mercury  placed  immediately  beneath  the  instrument,  and 
turning  the  whole  instrument  about  the  axis  until  the  middle  wire 
of  the  reticle  seen  directly  exactly  coincides  with  the  image  of 
this  wire  seen  by  reflection  from  the  surface  of  the  quicksilver. 
When  this  is  the  case,  the  telescope  is  vertical,  as  can  be  easily 
seen,  and  the  nadir  reading  may  be  found  from  the  circles. 
The  meridian  circle  thus  serves  to  determine  both  the  right  ascen- 
sion and  declination  of  a  given  star  at  the  same  culmination.  Zone 
observations  are  made  with  it  by  clamping  the  telescope  in  one 


ASTRONOMY. 


Fifc.l. 


Fig.3. 


T.g.4. 


FlG.    36. — BEADING   MICROSCOPE,    MICROMETER  AND   LEVEL. 


THE  EQUATORIAL.  87 

direction,  and  observing  successively  the  stars  which  pass  through 
its  field  of  view.  It  is  by  this  rapid  method  of  observing  that  the 
largest  catalogues  of  stars  have  been  formed. 

§  7.    THE  EQUATORIAL. 

To  complete  the  enumeration  and  description  of  the 
principal  instruments  of  astronomy,  we  require  an  account 
of  the  equatorial.  This  term,  properly  speaking,  refers 
to  a  form  of  mounting,  but  it  is  commonly  used  to  in- 
clude both  mounting  and  telescope.  In  this  class  of 
instruments  the  object  to  be  attained  is  in  general  the 
easy  finding  and  following  of  any  celestial  object  whose 
apparent  place  in  the  heavens  is  known  by  its  right  as- 
cension and  declination.  The  equatorial  mounting  con- 
sists essentially  of  a  pair  of  axes  at  right  angles  to  each 
other.  One  of  these  8  N  (the  polar  axis)  is  directed  to- 
ward the  elevated  pole  of  the  heavens,  and  it  therefore 
makes  an  angle  with  the  horizon  equal  to  the  latitude  of 
the  place  (p.  21).  This  axis  can  be  turned  about  its  own 
axial  line.  On  one  extremity  it  carries  another  axis  L  D 
(the  declination  axis},  which  is  fixed  at  right  angles  to  it, 
but  which  can  again  be  rotated  about  its  axial  line. 

To  this  last  axis  a  telescope  is  attached,  which  may 
either  be  a  reflector  or  a  refractor.  It  is  plain  that  such  a 
telescope  may  be  directed  to  any  point  of  the  heavens  ; 
for  we  can  rotate  the  declination  axis  until  the  telescope 
points  to  any  given  polar  distance  or  declination.  Then, 
keeping  the  telescope  fixed  in  respect  to  the  declination 
axis,  we  can  rotate  the  whole  instrument  as  one  mass 
about  the  polar  axis  until  the  telescope  points  to  any  por- 
tion of  the  parallel  of  declination  defined  by  the  given 
right  ascension  or  hour-angle.  Fig.  37  is  an  equatorial  of 
six-inch  aperture  which  can  be  moved  from  place  to  place. 

If  we  point  such  a  telescope  to  a  star  when  it  is  rising 
(doing  this  by  rotating  the  telescope  first  about  its  decli- 
nation axis,  and  then  about  the  polar  axis),  and  fix  the 
telescope  in  this  position,  we  can,  by  simply  rotating  the 


ASTRONOMY, 


FlG.  37. — EQUATORIAL  TELESCOPE  POINTED  TOWAltD  THE   POLE. 


THE  MICROMETER.  89 

whole  apparatus  on  the  polar  axis,  cause  the  telescope  to 
trace  out  on  the  celestial  sphere  the  apparent  diurnal  path 
which  this  star  will  appear  to  follow  from  rising  to  set- 
ting. In  such  telescopes  a  driving-clock  is  so  arranged 
that  it  can  turn  the  telescope  round  the  polar  axis  at  the 
same  rate  at  which  the  earth  itself  turns  about  its  own  axis 
of  rotation,  but  in  a  contrary  direction.  Hence  such  a 
telescope  once  pointed  at  a  star  will  continue  to  point  at  it 
as  long  as  the  driving-clock  is  in  operation,  thus  enabling 
the  astronomer  to  observe  it  at  his  leisure. 


FlG.    38. — MEASUREMENT  OP  POSITION- ANGLE. 

Every  equatorial  telescope  intended  for  making  exact  measures 
has  a  filar  micrometer,  which  is  precisely  the  same  in  principle  as 
the  reading  microscope  in  Fig.  2,  page  86,  except  that  its  two  wires 
are  parallel. 

A  figure  of  this  instrument  is  given  in  Fig.  3,  page  86.  One  of 
the  wires  is  fixed  and  the  other  is  movable  by  the  screw.  To 
measure  the  distance  apart,  of  two  objects  A  and  B,  wire  1  (the 
fixed  wire)  is  placed  on  A  and  wire  2  (movable  by  the  screw)  is 
placed  on  B.  The  number  of  revolutions  and  parts  of  a  revolution 
of  the  screw  is  noted,  say  10r-267  ;  then  wires  I  and  2  are  placed 
in  coincidence,  and  this  zero-reading  noted,  say  5r-143.  The  dis- 
tance A  B  is  equal  to  5r- 124.  Placing  wires  1  and  2  a  known  num- 
ber of  revolutions  apart,  we  may  observe  the  transits  of  a  star  in  the 
equator  over  them  ;  and  from  the  interval  of  time  required  for  this 
star  to  move  over  say  fifty  revolutions,  the  value  of  one  revolution 


90  ASTRONOMY. 

is  known,  and  can  always  be  used  to  turn  distances  measured  in 
revolutions  to  distances  in  time  or  arc. 

By  the  filar  micrometer  we  can  determine  the  distance  apart  in 
seconds  of  arc  of  any  two  stars  A  and  B.  To  completely  fix  the 
relative  position  of  A  and  B,  we  require  not  only  this  distance,  but 
also  the  angle  which  the  line  A  B  makes  with  some  fixed  direction 
in  space.  We  assume  as  the  fixed  direction  that  of  the  meridian 
passing  through  A.  Suppose  in  Fig.  38  A  and  B  to  be  two 
stars  visible  in  the  field  of  the  equatorial.  The  clock-work 
is  detached,  and  by  the  diurnal  motion  of  the  earth  the  two 
stars  will  cross  the  field  slowly  in  the  direction  of  the  parallel  of 
declination  passing  through  A,  or  in  the  direction  of  the  arrow  in 
the  figure  from  E.  to  W.,  east  to  west.  The  filar  micrometer  is  con- 
structed so  that  it  can  be  rotated  bodily  about  the  axis  of  the  tele- 
scope, and  a  graduated  circle  measures  the  amount  of  this  rotation. 
The  micrometer  is  then  rotated  until  the  star  A  will  pass  along 
one  of  its  wires.  This  wire  marks  the  direction  of  the  parallel. 
The  wire  perpendicular  to  this  is  then  in  the  meridian  of  the  star. 

The  position  angle  of  B  with  respect  to  A  is  then  the  angle  which 
A  B  makes  with  the  meridian  A  N  passing  through  A  toward  the 
north.  It  is  zero  when  B  is  north  of  A,  90°  when  B  is  east,  180 
when  B  is  south,  and  270°  when  B  is  west  of  A.  Knowing  p,  the 
position  angle  (N  A  B  in  the  figure),  and  *  (A  B)  the  distance  of  B, 
we  can  find  the  difference  of  right  ascension  (A  <*),  and  the  differ- 
ence of  declination  (A<J)  of  B  from  A  by  the  formulae, 

Aa  =  s  sin  p  ;   A(5  =  s  cos  p. 

Conversely  knowing  A«  and  Ad,  we  can  deduce  s  and  p  from 
these  formulae.  The  angle  p  is  measured  while  the  clock-work 
keeps  the  star  A  in  the  centre  of  the  field. 

§  8.    THE    ZENITH    TELESCOPE. 

The  accompanying  figure  gives  a  view  of  the  zenith  telescope  in 
the  form  in  which  it  is  used  by  the  United  States  Coast  Survey.  It 
consists  of  a  vertical  pillar  which  supports  two  Ts.  In  these 
rests  the  horizontal  axis  of  the  instrument  which  carries  the  tele- 
scope at  one  end,  and  a  counterpoise  at  the  other.  The  whole  in- 
strument can  revolve  180°  in  azimuth  about  this  pillar.  The  tele- 
scope has  a  micrometer  at  its  eye-end,  and  it  also  carries  a  divided 
circle,  provided  with  a  fine  level.  A  second  level  is  provided, 
whose  use  is  to  make  the  rotation  axis  horizontal.  The  peculiar 
features  of  the  zenith  telescope  are  the  divided  circle  and  its  at- 
tached level.  The  level  is,  as  shown  in  the  cut,  in  the  plane  of 
motion  of  the  telescope  (usually  the  plane  of  the  meridian),  and  it 
can  be  independently  rotated  on  the  axis  of  the  divided  circle,  and 
set  by  means  of  it  to  any  angle  with  the  optical  axis  of  the  telescope. 
The  circle  is  divided  from  zero  (0?)  at  its  lowest  point  to  90°  in 
each  direction,  and  is  firmly  attached  to  the  telescope  tube,  and 
moves  with  it. 

By  setting  the  vernier  or  index-arm  of  the  circle  to  any  degree 
and  minute  as  «,  and  clamping  it  there  (the  level  moving  with  it), 


THE  ZENITH  TELESCOPE.  91 


FlG.   39. — THE   ZENITH  TELESCOPE. 


92  ASTRONOMY. 

and  then  rotating  the  telescope  and  the  whole  system  about  the 
horizontal  axis  until  the  bubble  of  the  level  is  in  the  centre  of  the 
level-tube,  the  axis  of  the  telescopes  will  be  directed  to  the  zenith 
distance  a.  The  filar  micrometer  is  so  adjusted  that  a  motion  of  its 
screw  measures  differences  of  zenith  distance.  The  use  of  the  ze- 
nith telescope  is  for  determining  the  latitude  by  TALCOTT'S 
method.  The  theory  of  this  operation  has  been  already  given  on 
page  48.  A  description  of  the  actual  process  of  observation  will 
illustrate  the  excellences  of  this  method. 

Two  stars,  A  and  B,  are  selected  beforehand  (from  Star  Cata- 
logues), which  culminate,  A  south  of  the  zenith  of  the  place  of  ob- 
servation, B  north  of  it.  They  are  chosen  at  nearly  equal  zenith  dis- 
tances £A  and  £B,  and  so  that  £A — £B  is  less  than  the  breadth  of  the 
field  of  view.  Their  right  ascensions  are  also  chosen  so  as  to  be  about 
the  same.  The  circle  is  then  set  to  the  mean  zenith  distance  of  the 
two  stars,  and  the  telescope  is  pointed  so  that  the  bubble  is  nearly  in 
the  middle  of  the  level.  Suppose  the  right  ascension  of  A  is  the 
smaller,  it  will  then  culminate  first.  The  telescope  is  then  turned 
to  the  south.  As  A  passes  near  the  centre  of  the  field  its  distance 
from  the  centre  is  measured  by  the  micrometer.  The  level  and 
micrometer  are  read,  the  whole  instrument  is  revolved  180°,  and 
star  B  is  observed  in  the  same  way. 

By  these  operations  we  have  determined  the  difference  of  the 
zenith  distances  of  two  stars  whose  declinations  6*  and  <5»  are 
known.  But  0  being  the  latitude, 

0  =  6A  +  £A  and  0  =  <5B  —  £B,  whence 

0  =  I  (<JA  +  0B)  +  \  (?  -  ? ). 

The  first  term  of  this  is  known  ;  the  second  is  measured  ;  so  that 
each  pair  of  stars  so  observed  gives  a  value  of  the  latitude  which 
depends  on  the  measure  of  a  very  small  arc  with  the  micrometer, 
and  as  this  arc  can  be  measured  with  great  precision,  the  exactness 
of  the  determination  of  the  latitude  is  equally  great. 

§  9.    THE   SEXTANT. 

The  sextant  is  a  portable  instrument  by  which  the  altitudes 
of  celestial  bodies  or  the  angular  distances  between  them  may 
be  measured.  It  is  used  chiefly  by  navigators  for  determining  the 
latitude  and  the  local  time  of  the  position  of  the  ship.  Knowing 
the  local  time,  and  comparing  it  with  a  chronometer  regulated  on 
Greenwich  time,  the  longitude  becomes  known  and  the  ship's  place 
is  fixed. 

It  consists  of  the  arc  of  a  divided  circle  usually  60°  in  extent, 
whence  the  name.  This  arc  is  in  fact  divided  into  120  equal  parts, 
each  marked  as  a  degree,  and  these  are  again  divided  into  smaller 
spaces,  so  that  by  means  of  the  vernier  at  the  end  of  the  index-arm 
M  8  an  arc  of  10"  (usually)  may  be  read. 

The  index-arm  M  8  carries  the  index-glass  M,  which  is  a  silvered 
plane  mirror  set  perpendicular  to  the  plane  of  the  divided  arc.  The 


THE  SEXTANT. 


93 


horizon-glass  m  is  also  a  plane  mirror  fixed   perpendicular  to  the 
plane  of  the  divided  circle. 

This  last  glass  is  fixed  in  position,  while  the  first  revolves  with 
the  index-arm.  The  horizon-glass  is  divided  into  two  parts,  of 
which  the  lower  one  is  silvered,  the  upper  half  being  transparent. 
E  is  a  telescope  of  low  power  pointed  toward  the  horizon-glass. 
By  it  any  object  to  which  it  is  directed  can  be  seen  through  the  un- 
silvered  half  of  the  horizon-glass.  Any  other  object  in  the  same 
plane  can  be  brought  into  the  same  field  by  rotating  the  index-arm 


FlG.    40.— THE  SEXTANT. 

(and  the  index-glass  with  it),  so  that  a  beam  of  light  from  this 
second  object  shall  strike  the  index-glass  at  the  proper  angle,  there 
to  be  reflected  to  the  horizon-glass,  and  again  reflected  down  the 
telescope  E.  Thus  the  images  of  any  two  objects  in  the  plane  of 
the  sextant  may  be  brought  together  in  the  telescope  by  viewing 
one  directly,  and  the  other  by  reflection. 

The  principle  upon  which  the  sextant  depends  is  the  following, 
which  is  proved  in  optical  works.  The  angle  between  the  first  and 
the  last  direction  of  a  ray  which  has  suffered  two  rejections  in  the  same 


04  ASTRONOMY. 

plane  is  equal  to  twice  the  angle  which  the  two  reflecting  surfaces  make 
with  each  other. 

In  the  figure  S  A  is  the  ray  incident  upon  A,  and  this  ray  is  by 
reflection  brought  to  the  direction  B  E.  The  theorem  declares 
that  the  angle  B  E  S  is  equal  to  twice  D  C  J5,  or  twice  the  angle  of 


FIG. 


the  mirrors,  since  B  G  and  D  C  are  perpendicular  to  B  and  A.  To 
measure  the  altitude  of  a  star  (or  the  sun)  at  sea,  the  sextant  is  held 
in  the  hand,  and  the  telescope  is  pointed  to  the  sea-horizon,  which 
appears  like  a  definite  line.  The  index-arm  is  then  moved  until 
the  reflected  image  of  the  sun  or  of  the  star  coincides  with  the 


PlG.    42. — ARTIFICIAL   HORIZON. 

image  of  the  sea-horizon  seen  directly.  When  this  occurs  the  time 
is  to  be  noted  from  a  chronometer.  If  a  star  is  observed,  the  read- 
ing of  the  divided  limb  gives  the  altitude  directly  ;  if  it  is  the 
sun  or  moon  which  has  been  observed,  the  lower  limb  of  these  is 
brought  to  coincide  with  the  horizon,  and  the  altitude  of  the  centre 


THE  SEXTANT.  95 

is  found  by  applying  the  semi-diameter  as  found  in  the  Nautical 
Almanac  to  the  observed  altitude  of  the  limb. 

The  angular  distance  apart  of  a  star  and  the  moon  can  be  meas- 
ured by  pointing  the  telescope  at  the  star,  revolving  the  whole  sex- 
tant about  the  sight-line  of  the  telescope  until  the  plane  of  the  di- 
vided arc  passes  through  both  star  and  moon,  and  then  by  moving 
the  index-arm  until  the  reflected  moon  is  just  in  contact  with  the 
star's  image  seen  directly. 

On  shore  the  horizon  is  broken  up  by  buildings,  trees,  etc.,  and 
the  observer  is  therefore  obliged  to  have  recourse  to  an  artificial 
horizon,  which  consists  usually  of  the  reflecting  surface  of  some 
liquid,  as  mercury,  contained  in  a  small  vessel  A,  whose  upper 
surface  is  necessarily  parallel  to  the  horizon  D  A  C.  A  ray  of  light 
8  A,  from  a  star  at  S,  incident  on  the  mercury  at  A,  will  be  reflected 
in  the  direction  A  E,  making  the  angle  8  A  C  =  C  A  S'  (A  8'  be- 
ing E  A  produced),  and  the  reflected  image  of  the  star  will  appear 
to  an  eye  at  E  as  far  below  the  horizon  as  the  real  star  is  above  it. 
With  a  sextant  whose  index  and  horizon-glasses  are  at  /and  //,  the 
angle  8  E  8'  may  be  measured  ;  but  8  E  8'  =  8  A  S'  —  A  8  E, 
and  if  A  Eis  exceedingly  small  as  compared  with  A  S,  as  it  is  for 
all  celestial  bodies,  the  angle  A  8  E  may  be  neglected,  and  8  E  S' 
will  equal  8  A  S',  or  double  the  altitude  of  the  object :  hence  one 
half  the  reading  of  the  instrument  will  give  the  apparent  altitude. 


CHAPTER  III. 

MOTION    OF    THE    EARTH. 

§  1.    ANCIENT  IDEAS  OP  THE  PLANETS. 

IT  was  observed  by  the  ancients  that  while  the  great 
mass  of  the  stars  maintained  their  positions  relatively  to 
each  other  not  only  during  each  diurnal  revolution,  but 
month  after  month  and  year  after  year,  there  were  visi- 
ble to  them  seven  heavenly  bodies  which  changed  their 
positions  relatively  to  the  stars  and  to  each  other.  These 
they  called  planets  or  wandering  stars.  Still  calling  the 
apparent  crystalline  vault  in  which  the  stars  seem  to 
be  set  the  celestial  sphere,  and  imagining  it  as  at  rest, 
it  was  found  that  the  seven  planets  performed  a  very 
slow  revolution  around  the  sphere  from  west  to  east, 
in  periods  ranging  from  one  month  in  the  case  of  the 
moon,  to  thirty  years  in  that  of  Saturn.  It  was  evident 
that  these  bodies  could  not  be  considered  as  set  in  the 
same  solid  sphere  with  the  stars,  because  they  could  not 
then  change  their  positions  among  the  stars.  Various 
ways  of  accounting  for  their  motions  were  therefore  pro- 
posed. One  of  the  earliest  conceptions  is  associated  with 
the  name  of  PYTHAGORAS.  He  is  said  to  have  taught  that 
each  of  the  seven  planets  had  its  own  sphere  inside  of  and 
concentric  with  that  of  the  fixed  stars,  and  that  these 
seven  hollow  spheres  each  performed  its  own  revolution, 
independently  of  the  others.  This  idea  of  a  number  of  con- 
centric solid  spheres  was,  however,  apparently  given  up 


THE  SOLAR  SYSTEM.  97 

without  any  one  having  taken  the  trouble  to  refute  it  by 
argument.  Although  at  first  sight  plausible  enough,  a 
close  examination  would  show  it  to  be  entirely  inconsis- 
tent with  the  observed  facts.  The  idea  of  the  fixed  stars 
being  set  in  a  solid  sphere  was,  indeed,  in  seemingly 
perfect  accord  with  their  diurnal  revolution  as  observed 
by  the  naked  eye.  But  it  was  not  so  with  the  planets. 
The  latter,  after  continued  observation,  were  found  to 
move  sometimes  backward  and  sometimes  forward  ;  and 
it  was  quite  evident  that  at  certain  periods  they  were 
nearer  the  earth  than  at  other  periods.  These  motions 
were  entirely  inconsistent  with  the  theory  that  they  were 
fixed  in  solid  spheres.  Still  the  old  language  continued  in 
use — the  word  sphere  meaning,  not  a  solid  body,  but  the 
space  or  region  within  which  the  planet  moved. 

These  several  conceptions,  as  well  as  those  which  fol- 
lowed, were  all  steps  toward  the  truth.  The  planets  were 
rightly  considered  as  bodies  nearer  to  us  than  the  fixed 
stars.  It  was  also  rightly  judged  that  those  which  moved 
most  slowly  were  the  most  distant,  and  thus  their  order  of 
distance  from  the  earth  was  correctly  given,  except  in  the 
case  of  Mercury  and  Venus. 

We  now  know  that  these  seven  planets,  together  with 
the  earth,  and  a  number  of  other  bodies  which  the  tele- 
scope has  made  known  to  us,  form  a  family  or  system  by 
themselves,  the  dimensions  of  which,  although  inconceiv- 
ably greater  than  any  which  we  have  to  deal  with  at  the 
surface  of  the  earth,  are  quite  insignificant  when  com- 
pared with  the  distance  which  separates  us  from  the  fixed 
stars.  The  sun  being  the  great  central  body  of  this  sys- 
tem, it  is  called  the  Solar  System.  It  is  to  the  motions  of 
its  several  bodies  and  the  consequences  which  flow  from 
them  that  the  attention  of  the  reader  is  directed  in  the 
following  chapters.  We  premise  that  there  are  now  known 
to  be  eight  large  planets,  of  which  the  earth  is  the  third 
in  the  order  of  distance  from  the  sun,  and  that  these 
bodies  all  perform  a  regular  revolution  around  the  sun. 


98  ASTRONOMY. 

Mercury,  the  nearest,  performs  its  revolution  in  three 
months  ;  Neptune,  the  farthest,  in  164  years. 

First  in  importance  to  us,  among  the  heavenly  bodies 
which  we  see  from  the  earth,  stands  the  sun,  the  supporter 
of  life  and  motion  upon  the  earth.  At  first  sight  it  might 
seem  curious  that  the  sun  and  seeming  stars  like  Mars 
and  Saturn  should  have  been  classified  together  as  planets 
by  the  ancients,  while  the  fixed  stars  were  considered  as 
forming  another  class.  That  the  ancients  were  acute 
enough  to  do  this  tends  to  impress  us  with  a  favorable 
sense  of  the  scientific  character  of  their  intellect.  To  any 
but  the  most  careful  theorists  and  observers,  the  star-like 
planets,  if  we  may  call  them  so,  would  never  have  seemed 
to  belong  in  the  same  class  with  the  sun,  but  rather  in 
that  of  the  stars  ;  especially  when  it  was  found  that  they 
were  never  visible  at  the  same  time  with  the  sun.  But 
before  the  times  of  which  we  have  any  historic  record, 
there  were  men  who  saw  that,  in  a  motion  from  west  to 
east  among  the  fixed  stars,  these  several  bodies  showed  a 
common  character,  which  was  more  essential  in  a  theory 
of  the  universe  than  were  their  immense  differences  of 
aspect  and  lustre,  striking  though  these  might  be. 

It  must,  however,  be  remembered  that  we  no  longer 
consider  the  sun  as  a  planet.  We  have  modified  the  an- 
cient system  by  making  the  sun  and  the  earth  change 
places,  so  that  the  latter  is  now  regarded  as  one  of  the  eight 
large  planets,  while  the  former  has  taken  the  place  of  the 
earth  as  the  central  body  of  the  system.  In  consequence 
of  the  revolution  of  the  planets  round  the  sun,  each  of 
them  seems  to  perform  a  corresponding  circuit  in  the 
heavens  around  the  celestial  sphere,  when  viewed  from 
any  other  planet  or  from  the  earth. 

§  2.    ANNUAL  KEVOLUTION   OP  THE  EARTH. 

To  an  observer  on  the  earth,  the  sun  seems  to  perform  an 
annual  revolution  among  the  stars,  a  fact  which  has  been 
known  from  the  earliest  ages.  We  now  know  that  this 


MOTION  OF  THE  EARTH.  09 

is  due  to  the  annual  revolution  of  the  earth  round  the 
sun.  It  is  to  the  nature  and  effects  of  this  annual  revolu- 
tion of  the  earth  that  the  attention  of  the  reader  is  now 
directed.  Our  first  lesson  is  to  show  the  relations  between 
it  and  the  corresponding  apparent  revolution  of  the  sun, 
which  is  its  counterpart. 

In  Fig.  43,  let  S  represent  the  sun,  A  B  CD  the  orbit 
of  the  earth  around  it,  and  EFG II  the  sphere  of  the 


FlG.    43. — REVOLUTION   OF   THE   EARTH. 

fixed  stars.  This  sphere,  being  supposed  infinitely  dis- 
tant, must  be  considered  as  infinitely  larger  than  the  circle 
A  B  G  D.  Suppose  now  that  1,  2,  3,  4,  5,  6  are  a 
number  of  consecutive  positions  of  the  earth.  The  line 
1$  drawn  from  the  sun  to  the  earth  in  the  first  position  is 
called  the  radius  vector  of  the  earth.  Suppose  this  line 
extended  infinitely  so  as  to  meet  the  celestial  sphere  in 
the  point  V.  It  is  evident  that  to  an  observer  on  the 


100  ASTRONOMY. 

eartli  at  1  the  sun  will  appear  projected  on  the  sphere 
in  the  direction  of  1'.  When  the  earth  reaches  2,  it 
will  appear  in  the  direction  of  2',  and  so  on.  In  other 
words,  as  the  earth  revolves  around  the  sun,  the  latter 
will  seem  to  perform  a  revolution  among  the  fixed  stars, 
which  are  immensely  more  distant  than  itself. 

It  is  also  evident  that  the  point  in  which  the  earth  would 
be  projected,  if  viewed  from  the  sun,  is  always  exactly 
opposite  that  in  which  the  sun  appears  as  projected  from 
the  earth.  Moreover,  if  the  earth  moves  more  rapidly  in 
some  points  of  its  orbit  than  in  others,  it  is  evident  that 
the  sun  will  also  appear  to  move  more  rapidly  among  the 
stars,  and  that  the  two  motions  must  always  accurately 
correspond  to  each  other. 

The  radius  vector  of  the  earth  in  its  annual  course  de- 
scribes a  plane,  which  in  the  figure  may  be  represented  by 
that  of  the  paper.  This  plane  continued  to  infinity  in 
every  direction  will  cut  the  celestial  sphere  in  a  great  cir- 
cle ;  and  it  is  evident  that  the  sun  will  always  appear  to 
move  in  this  circle.  The  plane  and  the  circle  are  indiffer- 
ently termed  the  ecliptic.  The  plane  of  the  ecliptic  is 
generally  taken  as  the  fundamental  one,  to  which  the  po- 
sitions of  all  the  bodies  in  the  solar  system  are  referred. 
By  the  fundamental  principles  of  spherical  trigonometry,  it 
divides  the  celestial  sphere  into  two  equal  parts.  In  think- 
ing of  the  celestial  motions,  it  is  convenient  to  conceive  of 
this  plane  as  horizontal.  Then  if  we  draw  a  vertical  line 
passing  through  the  sun  at  right  angles  to  it,  or  perpen- 
dicular to  the  plane  of  the  paper  on  which  the  figure  is 
represented,  the  point  at  which  this  line  intersects  the 
celestial  sphere  will  be  the  pole  of  the  ecliptic.  This 
point  is  situated  in  the  constellation  Draco,  and  has  an  ex- 
tremely slow  motion  of  about  half  a  second  a  year,  owing 
to  a  change  in  the  position  of  the  ecliptic  to  be  hereafter 
described. 

Let  us  now  study  the  apparent  annual  revolution  of  the 
sun  produced  in  the  way  just  mentioned.  One  result  of 


THE  SUN'S  APPARENT *>  .PATH.,  / 

this  motion  is  probably  familiar  to  every  reader,  in  the 
different  constellations  which  are  seen  at  different  times  of 
the  year.  Let  us  take,  for  example,  the  bright  star  Alde- 
baran,  which,  on  a  winter  evening,  we  may  see  north- 
west of  Orion.  Near  the  end  of  February  this  star  crosses 
the  meridian  about  six  o'clock  in  the  evening,  and  sets 
about  midnight.  If  we  watch  it  night  after  night  through 
the  months  of  March  and  April,  we  shall  find  that  it  is  far- 
ther and  farther  toward  the  west  on  each  successive  even- 
ing at  the  same  hour.  By  the  end  of  April  we  shall  bare- 
ly be  able  to  see  it  about  the  close  of  the  evening  twilight. 
At  the  end  of  May  it  will  be  so  close  to  the  sun  as  to  be 
entirely  invisible.  This  shows  that  during  the  months  we 
have  been  watching  it,  the  sun  has  been  approaching  the 
star  from  the  west.  If  in  July  we  watch  the  eastern 
horizon  in  the  early  morning,  we  shall  see  this  star  rising 
before  the  sun.  The  sun  has  therefore  passed  by  the 
star,  and  is  now  east  of  it.  At  the  end  of  November  we 
will  find  it  rising  at  sunset  and  setting  at  sunrise.  The 
sun  is  therefore  directly  opposite  the  star.  During  the 
winter  months  it  approaches  it  again  from  the  west,  and 
passes  it  about  the  end  of  May,  as  before.  Any  other 
star  south  of  the  zenith  shows  a  similar  change,  since  the 
relative  positions  of  the  stars  do  not  vary. 

§  3.  THE  SUN'S  APPARENT  PATH. 

It  is  evident  that  if  the  apparent  path  of  the  sun  lay  in 
the  equator,  it  would,  during  the  entire  year,  rise  exactly 
in  the  east  and  set  in  the  west,  and  would  always  cross 
the  meridian  at  the  same  altitude.  The  days  would 
always  be  twelve  hours  long,  for  the  same  reason  that  a 
star  in  the  equator  is  always  twelve  hours  above  the  hori- 
zon and  twelve  hours  below  it.  But  we  know  that  this 
is  not  the  case,  the  sun  being  sometimes  north  of  the 
equator  and  sometimes  south  of  it,  and  therefore  having 
a  motion  in  declination.  To  understand  this  motion, 


102   J  l  ^  ASTRONOMY. 

suppose  that  on  March  19th,  1879,  the  sun  had  been 
observed  with  a  meridian  circle  and  a  sidereal  clock  at  the 
moment  of  transit  over  the  meridian  of  Washington.  Its 
position  would  have  been  found  to  be  this  : 

Eight  Ascension,  23h  55ra  238  ;  Declination,  0°  30'  south. 

Had  the  observation   been  repeated  on  the  20th  and 
following  days,  the  results  would  have  been  : 

March  20,  E.  Ascen.  23h  59ra    2s ;  Dec.  0°    6'  South. 

21,  "  Oh    2m40s;     "     0°  17' North. 

22,  "  Oh    6m  19s ;      "     0°  41'  North. 


FlG.    44.— THE   SUN  CROSSING  THE  EQUATOR. 

If  we  lay  these  positions  down  on  a  chart,  we  shall  find 
them  to  be  as  in  Fig.  44,  the  centre  of  the  sun  being 
south  of  the  equator  in  the  first  two  positions,  and  north 
of  it  in  the  last  two.  Joining  the  successive  positions  by 
a  line,  we  shall  have  a  small  portion  of  the  apparent  path 
of  the  sun  on  the  celestial  sphere,  or,  in  other  words,  a 
small  part  of  the  ecliptic. 

It  is  clear  from  the  observations  and  the  figure  that  the 
sun  crossed  the  equator  between  six  and  seven  o'clock  on 
the  afternoon  of  March  20th,  and  therefore  that  the  equa- 
tor and  ecliptic  intersect  at  the  point  where  the  sun  was  at 
that  hour.  This  point  is  called  the  vernal  equinox,  the 


THE  SUN'S  APPARENT  PATH. 


103 


first  word  indicating  the  season,  while  the  second 
expresses  the  equality  of  the 
nights  and  days  which  occurs 
when  the  sun  is  on  the  equator. 
It  will  be  remembered  that  this 
equinox  is  the  point  from  which 
right  ascensions  are  counted  in 
the  heavens  in  the  same  way 
that  longitudes  on  the  earth  are 
counted  from  Greenwich  or 
Washington.  The  sidereal 
clock  is  therefore  so  set  that 
the  hands  shall  read  0  hours 
0  minutes  0  seconds  at  the 
moment  when  the  vernal  equi- 
nox crosses  the  meridian. 

Continuing  our  observations 
of  the  sun's  apparent  course  for 
six  months  from  March  20th 
till  September  23d,  we  should 
find  it  to  be  as  in  Fig.  45.  It 
will  be  seen  that  Fig.  44  cor- 
responds to  the  right-hand  end 
of  45,  but  is  on  a  much  larger  * 
scale.  The  sun,  moving  along 
the  great  circle  of  the  ecliptic, 
will  reach  its  greatest  northern 
declination  about  June  21st. 
This  point  is  indicated  on  the 
figure  as  90°  from  the  vernal 
equinox,  and  is  called  the  sum- 
mer solstice.  The  sun's  right 
ascension  is  then  six  hours,  and 
its  declination  23J°  north. 

The  course  of  the  sun  now 
inclines  toward  the  south,  and 
it  again  crosses  the  equator  about  September  22d  at 


104  ASTRONOMY. 

a  point  diametrically  opposite  the  vernal  equinox.  In 
virtue  of  the  theorem  of  spherical  trigonometry  that  all 
great  circles  intersect  each  other  in  two  opposite  points, 
the  ecliptic  and  equator  intersect  at  the  two  opposite  equi- 
noxes. The  equinox  which  the  sun  crosses  on  September 
22d  is  called  the  autumnal  equinox. 

During  the  six  months  from  September  to  March  the 
sun's  course  is  a  counterpart  of  that  from  March  to  Sep- 
tember, except  that  it  lies  south  of  the  equator.  It  at- 
tains its  greatest  south  declination  about  December  22d, 
in  right  ascension  18  hours,  and  south  declination  23£°. 
This  point  is  called  the  winter  solstice.  It  then  begins  to 
incline  its  course  toward  the  north,  reaching  the  vernal 
equinox  again  on  March  20th,  1880. 

The  two  equinoxes  and  the  two  solstices  may  be  re- 
garded as  the  four  cardinal  points  of  the  sun's  apparent 
annual  circuit  around  the  heavens.  Its  passage  through 
these  points  is  determined  by  measuring  its  altitude  or 
declination  from  day  to  day  with  a  meridian  circle.  Since 
in  our  latitude  greater  altitudes  correspond  to  greater 
declinations,  it  follows  that  the  summer  solstice  occurs  on 
the  day  when  the  altitude  of  the  sun  is  greatest,  and  the 
winter  solstice  on  that  when  it  is  least.  The  mean  of 
these  altitudes  is  that  of  the  equator,  and  may  therefore 
be  found  by  subtracting  the  latitude  of  the  place  from 
90°.  The  time  when  the  sun  reaches  this  altitude  going 
north  marks  the  vernal  equinox,  and  that  when  it  reaches 
it  going  south  marks  the  autumnal  equinox. 

These  passages  of  the  sun  through  the  cardinal  points 
have  been  the  subjects  of  astronomical  observation  from 
the  earliest  ages  on  account  of  their  relations  to  the  change 
of  the  seasons.  An  ingenious  method  of  finding  the  time 
when  the  sun  reached  the  equinoxes  was  used  by  the  as- 
tronomers of  Alexandria  about  the  beginning  of  our  era. 
In  the  great  Alexandrian  Museum,  a  large  ring  or  wheel 
was  set  up  parallel  to  the  plane  of  the  equator — in  other 
words,  it  was  so  fixed  that  a  star  at  the  pole  would  shine 


THE  ZODIAC.  105 

perpendicularly  on  the  wheel.  Evidently  its  plane  if 
extended  must  have  passed  through  the  east  and  west 
points  of  the  horizon,  while  its  inclination  to  the  vertical 
was  equal  to  the  latitude  of  the  place,  which  was  not  "far 
from  30°.  When  the  sun  reached  the  equator  going  north 
or  south,  and  shone  upon  this  wheel,  its  lower  edge  would 
be  exactly  covered  by  the  shadow  of  the  upper  edge  ; 
whereas  in  any  other  position  the  sun  would  shine  upon 
the  lower  inner  edge.  Thus  the  time  at  which  the  sun 
reached  the  equinox  could  be  determined,  at  least  to  a 
fraction  of  a  day.  By  the  more  exact  methods  of  modern 
times,  it  can  be  determined  within  less  than  a  minute. 

It  will  be  seen  that  this  method  of  determining  the  an- 
nual apparent  course  of  the  sun  by  its  declination  or  alti- 
tude is  entirely  independent  of  its  relation  to  the  fixed 
stars ;  and  it  could  be  equally  well  applied  if  no  stars 
were  ever  visible.  There  are,  therefore,  two  entirely  dis- 
tinct ways  of  finding  when  the  sun  or  the  earth  has  com- 
pleted its  apparent  circuit  around  the  celestial  sphere  ; 
the  one  by  the  transit  instrument  and  sidereal  clock,  which 
show  when  the  sun  returns  to  the  same  position  among 
the  stars,  the  other  by  the  measurement  of  altitude,  which 
shows  when  it  returns  to  the  same  equinox.  By  the  for- 
mer method,  already  described,  we  conclude  that  it  has 
completed  an  annual  circuit  when  it  returns  to  the  same 
star  ;  by  the  latter  when  it  returns  to  the  same  equinox. 
These  two  methods  will  give  slightly  different  results  for 
the  length  of  the  year,  for  a  reason  to  be  hereafter 
described. 

The  Zodiac  and  its  Divisions. — The  zodiac  is  a  belt 
in  the  heavens,  commonly  considered  as  extending  some  8° 
on  each  side  of  the  ecliptic,  and  therefore  about  16°  wide. 
The  planets  known  to  the  ancients  are  always  seen  within 
this  belt.  At  a  very  early  age  the  zodiac  was  mapped  out 
into  twelve  signs  known  as  the  signs  of  the  zodiac,  the 
names  of  which  have  been  handed  down  to  the  present 
time.  Each  of  these  signs  was  supposed  to  be  the  seat  of 


106  ASTRONOMY. 

a  constellation  after  which  it  was  called.  Commencing 
at  the  vernal  equinox,  the  first  thirty  degrees  through 
which  the  sun  passed,  or  the  region  among  the  stars  in 
which  it  was  found  during  the  month  following,  was 
called  the  sign  Aries.  The  next  thirty  degrees  was  called 
Taurus.  The  names  of  all  the  twelve  signs  in  their 
proper  order,  with  the  approximate  time  of  the  sun's  en- 
tering upon  each,  are  as  follow  : 

Aries,  the  Ram,  March  20. 

Taurus,  the  Bull,  April  20. 

Gemini,  the  Twins,  May  20. 

Cancer,  the  Crab,  June  21. 

Leo,  the  Lion,  July  22. 

Virgo,  the  Virgin,  August  22. 

Libra,  the  Balance,  September  22. 

Scorpius,  the  Scorpion,  October  23. 

Sagittarius,  the  Archer,  November  23. 

Capricornus,  the  Goat,  December  21. 
Aquarius,  the  Water-bearer,        January  20. 

Pisces,  the  Fishes,  February  19. 

Each  of  these  signs  coincides  roughly  with  a  constella- 
tion in  the  heavens  ;  and  thus  there  are  twelve  constella- 
tions called  by  the  names  of  these  signs,  but  the  signs  and 
the  constellations  no  longer  correspond.  Although  the  sun 
now  crosses  the  equator  and  enters  the  sign  Aries  on  the 
20th  of  March,  he  does  not  reach  the  constellation  Aries 
until  nearly  a  month  later.  This  arises  from  the  preces- 
sion of  the  equinoxes,  to  be  explained  hereafter. 

§  4.    OBLIQUITY    OF    THE   ECLIPTIC. 

"We  have  already  stated  that  when  the  sun  is  at  the 
summer  solstice,  it  is  about  23^°  north  of  the  equator, 
and  when  at  the  winter  solstice,  about  23£°  south.  This 
shows  that  the  ecliptic  and  equator  make  an  angle 
of  about  23J°  with  each  other.  This  angle  is  called 


OBLIQUITY  OF  THE  ECLIPTIC.  107 

the  obliquity  of  the  ecliptic,  and  its  determination  is 
very  simple.  It  is  only  necessary  to  find  by  repeated 
observation  the  sun's  greatest  north  declination  at  the 
summer  solstice,  and  its  greatest  south  declination  at 
the  winter  solstice.  Either  of  these  declinations,  which 
must  be  equal  if  the  observations  are  accurately  made, 
will  give  the  obliquity  of  the  ecliptic.  It  has  been  con- 
tinually diminishing  from  the  earliest  ages  at  a  rate  of 
about  half  a  second  a  year,  or,  more  exactly,  about  forty- 
seven  seconds  in  a  century.  This  diminution  is  due  to 
the  gravitating  forces  of  the  planets,  and  will  continue  for 
several  thousand  years  to  come.  It  will  not,  however,  go 
on  indefinitely,  but  the  obliquity  will  only  oscillate  be- 
tween comparatively  narrow  limits. 

The  relation  of  the  obliquity  of  the  ecliptic  to  the  sea- 
sons is  quite  obvious.  When  the  sun  is  north  of  the  equa- 
tor, it  culminates  at  a  higher  altitude  in  the  northern  hem- 
isphere, and  more  than  half  of  its  apparent  diurnal  course 
is  above  the  horizon,  as  explained  in  the  chapter  on  the 
celestial  sphere.  Hence  we  have  the  heats  of  summer. 
In  the  southern  hemisphere,  of  course,  the  case  is  re- 
versed :  when  the  sun  is  in  north  declination,  less  than 
half  of  his  diurnal  course  is  above  the  horizon  in  that  hem- 
isphere. Therefore  this  situation  of  the  sun  corresponds 
to  summer  in  the  northern  hemisphere,  and  winter  in  the 
southern  one.  In  exactly  the  same  way,  when  the  sun  is 
far  south  of  the  equator,  the  days  are  shorter  in  the  north- 
ern hemisphere  and  longer  in  the  southern  hemisphere. 
It  is  therefore  winter  in  the  former  and  summer  in  the 
latter.  If  the  equator  and  the  ecliptic  coincided — that 
is,  if  the  sun  moved  along  the  equator — there  would 
be  no  such  thing  as  a  difference  of  seasons,  because  the 
sun  would  always  rise  exactly  in  the  east  and  set  exactly 
in  the  west,  and  always  culminate  at  the  same  altitude. 
The  days  would  always  be  twelve  hours  long  the  world 
over.  This  is  the  case  with  the  planet  Jupiter. 

In  the  preceding  paragraphs,   we  have  explained  the 


108  ASTRONOMY. 

apparent  annual  circuit  of  the  sun  relative  to  the  equator, 
and  shown  how  the  seasons  depend  upon  this  circuit.  In 
order  that  the  student  may  clearly  grasp  the  entire  subject, 
it  is  necessary  to  show  the  relation  of  these  apparent  move- 
ments to  the  actual  movement  of  the  earth  around  the 
sun. 

To  understand  the  relation  of  the  equator  to  the  eclip- 
tic, we  must  remember  that  the  celestial  pole  and  the 
celestial  equator  have  really  no  reference  whatever  to  the 
heavens,  but  depend  solely  on  the  direction  of  the  earth's 
axis  of  rotation.  The  pole  of  the  heavens  is  nothing 
more  than  that  point  of  the  celestial  sphere  toward  which 
the  earth's  axis  points.  If  the  direction  of  this  axis 
changes,  the  position  of  the  celestial  pole  among  the  stars 
will  change  also  ;  though  to  an  observer  on  the  earth, 
unconscious  of  the  change,  it  would  seem  as  if  the  starry 
sphere  moved  while  the  pole  remained  at  rest.  Again,  the 
celestial  equator  being  merely  the  great  circle  in  which  the 
plane  of  the  earth's  equator,  extended  out  to  infinity  in 
every  direction,  cuts  the  celestial  sphere,  any  change  in 
the  direction  of  the  pole  of  the  earth  necessarily  changes 
the  position  of  the  equator  among  the  stars.  Now  the 
positions  of  the  celestial  pole  and  the  celestial  equator 
among  the  stars  seem  to  remain  unchanged  throughout 
the  year.  (There  is,  indeed,  a  minute  change,  but  it  does 
not  affect  our  present  reasoning.)  This  shows  that,  as 
the  earth  revolves  around  the  sun,  its  axis  is  constantly 
directed  toward  nearly  the  same  point  of  the  celestial 
sphere. 

§   5.    THE  SEASONS. 

The  conclusions  to  which  we  are  thus  led  respecting 
the  real  revolution  of  the  earth  are  shown  in  Fig.  46. 
Here  8  represents  the  sun,  with  the  orbit  of  the  earth 
surrounding  it,  but  viewed  nearly  edgeways  so  as  to  be 
much  foreshortened.  A  B  CD  are  the  four  cardinal 
positions  of  the  earth  which  correspond  to  the  cardinal 


THE  SEASONS. 


109 


points  of  the  apparent  path  of  the  sun  already  described. 
In  each  figure  of  the  earth  N~  S  is  the  axis,  JV  being  its 
north  and  8  its  south  pole.  Since  this  axis  points  in  the 


FIG.  46. — CAUSES  OF  THE  SEASONS. 

same  direction  relative  to  the  stars  during  an  entire  year, 
it  follows  that  the  different  lines  N  S  are  all  parallel. 
Again,  since  the  equator  does  not  coincide  with  the  ecliptic, 
these  lines  are  not  perpendicular  to  the  ecliptic,  but  are 
inclined  from  this  perpendicular  by  23f°. 

Now,  consider  the  earth  as  at  A  ;  here  it  is  seen  that  the 
sun  shines  more  on  the  southern  hemisphere  than  on  the 
northern  ;  a  region  of  23-J0  around  the  north  pole  is  in 
darkness,  while  in  the  corresponding  region  around  the 
south  pole  the  sun  shines  all  day.  The  five  circles  at  right 
angles  to  the  earth's  axis  are  the  parallels  of  latitude  around 
which  each  region  on  the  surface  of  the  earth  is  carried  by 
the  diurnal  rotation  of  the  latter  on  its  axis.  It  will  be  seen 
that  in  the  northern  hemisphere  less  than  half  of  these  are 
illuminated  by  the  sun,  and  in  the  southern  hemisphere 
more  than  half.  This  corresponds  to  our  winter  solstice. 

When  the  earth  reaches  B,  its  axis  N  S\&  at  right  an- 
gles to  the  line  drawn  to  the  sun,  so  that  the  latter  shines 
perpendicularly  on  the  equator,  the  plane  of  which  passes 
through  it.  The  diurnal  circles  on  the  earth  are  one  half 


110  ASTRONOMY. 

illuminated  and  one  half  in  darkness.  This  position  cor- 
responds to  the  vernal  equinox. 

At  C  the  case  is  exactly  the  reverse  of  that  at  A,  the 
sun  shining  more  on  the  northern  hemisphere  than  on  the 
southern  one.  North  of  the  equator  more  than  half  the 
diurnal  circles  are  in  the  illuminated  hemisphere,  and  south 
of  it  less.  Here  then  we  have  winter  in  the  southern  and 
summer  in  the  northern  hemisphere.  The  sun  is  above  a 
region  23-J-0  north  of  the  equator,  so  that  this  position  cor- 
responds to  our  summer  solstice. 

At  D  the  earth's  axis  is  once  more  at  right  angles  to  a 
line  drawn  to  the  sun.  The  latter  therefore  shines  upon 
the  equator,  and  we  have  the  autumnal  equinox. 

In  whatever  position  we  suppose  the  earth,  the  line  SN, 
continued  indefinitely,  meets  the  celestial  sphere  at  its 
north  pole,  while  the  middle  or  equatorial  circle  of  the 
earth,  continued  indefinitely  in  every  direction,  marks  out 
the  celestial  equator  in  the  heavens.  At  first  sight  it  might 
seem  that,  owing  to  the  motion  of  the  earth  through  so 
vast  a  circuit,  the  positions  of  the  celestial  pole  and  equa- 
tor must  change  in  consequence  of  this  motion.  "We  might 
say  that,  in  reality,  the  pole  of  the  earth  describes  a  circle  in 
the  celestial  sphere  of  the  same  size  as  the  earth's  orbit. 
But  this  sphere  being  infinitely  distant,  the  circle  thus  de- 
scribed appears  to  us  as  a  point,  and  thus  the  pole  of  the 
heavens  seems  to  preserve  its  position  among  the  stars 
through  the  whole  course  of  the  year.  Again,  we  may 
suppose  the  equator  to  have  a  slight  annual  motion  among 
the  stars  from  the  same  cause.  But  for  the  same  reason 
this  motion  is  nothing  when  seen  from  the  earth.  On  the 
other  hand,  the  slightest  change  in  the  direction  of  the 
axis  S  N  will  change  the  apparent  position  of  the  pole 
among  the  stars  by  an  angle  equal  to  that  change  of  direc- 
tion. We  may  thus  consider  the  position  of  the  celestial 
pole  as  independent  of  the  position  of  the  earth  in  its 
orbit,  and  dependent  entirely  on  the  direction  in  which 
the  axis  of  the  earth  points. 


CELESTIAL  LATITUDE  AND  LONGITUDE.         Ill 

If  this  axis  were  perpendicular  to  the  plane  of  the  eclip- 
tic, it  is  evident  that  the  sun  would  always  lie  in  the  plane 
of  the  equator,  and  there  would  be  no  change  of  seasons 
except  such  slight  ones  as  might  result  from  the  small 
differences  in  the  distance  of  the  earth  at  different  seasons. 

§  6.    CELESTIAL  LATITUDE  AND  LONGITUDE. 

Besides  the  circles  of  reference  described  in  the  first 
chapter,  still  another  system  is  used  in  which  the  ecliptic 
is  taken  as  the  fundamental  plane.  Since  the  motion  of 
the  earth  around  the  sun  takes  place,  by  definition,  in  the 
plane  of  the  ecliptic,  and  the  motions  of  the  planets  very 
near  that  plane,  it  is  frequently  more  convenient  to  refer 
the  positions  of  the  planets  to  the  plane  of  the  ecliptic  than 
to  that  of  the  equator.  The  co-ordinates  of  a  heavenly 
body  thus  referred  are  called  its  celestial  latitude  and 
longitude.  To  show  the  relation  of  these  co-ordinates  to 
right  ascension  and  declination,  we  give  a  figure  showing 
both  co-ordinates  at  the  same  time,  as  marked  on  the 
celestial  sphere.  This  figure  is  supposed  to  be  the  celes- 
tial sphere,  having  the  solar  system  in  its  centre.  The 
direction  P  Q  is  that  of  the  axis  of  the  earth  ;  IJ\$>  the 
ecliptic,  or  the  great  circle  in  which  the  plane  of  the 
earth's  orbit  intersects  the  celestial  sphere.  The  point  in 
which  these  two  circles  cross  is  marked  Oh,  and  is  the  ver- 
nal equinox  from  which  the  right  ascension  and  the  longi- 
tude are  both  counted. 

The  horizontal  and  vertical  circles  show  how  right  ascen- 
sion and  declination  are  counted  in  the  manner  described  in 
Chapter  I.  As  the  right  ascension  is  counted  all  the  way 
around  the  equator  from  Oh  to  24h,  so  longitude  is  counted 
along  the  ecliptic  from  the  point  Oh,  or  the  vernal  equinox, 
toward  J  in  degrees.  The  whole  circuit  measuring  360°, 
this  distance  will  carry  us  all  the  way  round.  Thus  if  a 
body  lies  in  the  ecliptic,  its  longitude  is  simply  the  number 
of  degrees  from  the  vernal  equinox  to  its  position,  meas- 
ured in  the  direction  from  1  toward  J.  If  it  does  not  lie 


ASTRONOMY. 

in  the  ecliptic ;  if,  for  instance,  it  is  at  the  point  B,  we 
let  fall  a  perpendicular  B  J  from  the  body  upon  the 
ecliptic.  The  length  of  this  perpendicular,  measured  in 
degrees,  is  called  the  latitude  of  the  body,  which  may  be 
north  or  south,  while  the  distance  of  the  foot  of  the  per- 
pendicular from  the  vernal  equinox  is  called  its  longitude. 
In  astronomy  it  is  usual  to  represent  the  positions  of  the 
bodies  of  the  solar  system,  relatively  to  the  sun,  by  their 
longitudes  and  latitudes,  because  in  the  ecliptic  we  have  a 


FlG.    47. — CIRCLES  OF  THE   SPHERE. 

• 

plane  more  nearly  fixed  than  that  of  the  equator.  On  the 
other  hand,  it  is  more  convenient  to  represent  the  position 
of  all  the  heavenly  bodies  as  seen  from  the  earth  by  their 
right  ascensions  and  declinations,  because  we  cannot  meas- 
ure their  longitudes  and  latitudes  directly,  but  can  only 
observe  right  ascension  and  declination.  If  we  wish  to 
determine  the  longitude  and  latitude  of  a  body  as  seen 
from  the  centre  of  the  earth,  we  have  to  first  find  its  right 
ascension  and  declination  by  observation,  and  then  change 
these  quantities  to  longitude  and  latitude  by  trigonometri- 
cal formulae. 


CHAPTER    IV. 

THE  PLANETAKY  MOTIONS. 

§  1.  APPARENT   AND    REAL   MOTIONS    OF    THE 
PLANETS. 

Definitions. — The  solar  system,  as  we  now  know  it,  com- 
prises so  vast  a  number  of  bodies  of  various  orders  of  mag- 
nitude and  distance,  and  subjected  to  so  many  seemingly 
complex  motions,  that  we  must  consider  its  parts  sepa- 
rately. Our  attention  will  therefore,  in  the  present  chap- 
ter, be  particularly  directed  to  the  motions  of  the  great 
planets,  which  we  may  consider  as  forming,  in  some  sort, 
the  fundamental  bodies  of  the  system.  These  bodies 
may,  with  respect  to  their  apparent  motions,  be  divided 
into  three  classes. 

Speaking,  for  the  present,  of  the  sun  as  a  planet,  the 
first  class  comprises  the  sun  and  moon.  We  have  seen 
that  if,  upon  a  star  chart,  we  mark  down  the  positions  of 
the  sun  day  by  day,  they  will  all  fall  into  a  regular  circle 
which  marks  out  the  ecliptic.  The  monthly  course  of  the 
moon  is  found  to  be  of  the  same  nature,  although  its 
motion  is  by  no  means  uniform  in  a  month,  yet  it  is 
always  toward  the  east,  and  always  along  or  very  near  a 
certain  great  circle. 

The  second  class  comprises  Venus  and  Mercury.  The 
peculiarity  exhibited  by  the  apparent  motion  of  these 
bodies  is,  that  it  is  an  oscillating  one  on  each  side  of  the 
sun.  If  we  watch  for  the  appearance  of  one  of  these 
planets  after  sunset  from  evening  to  evening,  we  shall  lind 


114  ASTRONOMY. 

it  to  appear  above  the  western  horizon.  Night  after  night 
it  will  be  farther  and  farther  from  the  sun  until  it  attains 
a  certain  maximum  distance  ;  then  it  will  appear  to  return 
to  the  sun  again,  and  for  a  while  to  be  lost  in  its  rays. 
A  few  days  later  it  will  reappear  to  the  west  of  the  sun, 
and  thereafter  be  visible  in  the  eastern  horizon  before 
sunrise.  In  the  case  of  Mercury ,  the  time  required  for 
one  complete  oscillation  back  and  forth  is  about  four 
months  ;  and  in  the  case  of  Venus  more  than  a  year  and 
a  half. 

The  third  class  comprises  Mars,  Jupiter,  and  /Saturn  as 
well  as  a  great  number  of  planets  not  visible  to  the  naked 
eye.  The  general  or  average  motion  of  these  planets  is 
toward  the  east,  a  complete  revolution  in  the  celestial 
sphere  being  performed  in  times  ranging  from  two  years 
in  the  case  of  Mars  to  164  years'  in  that  of  Neptune. 
But,  instead  of  moving  uniformly  forward,  they  seem  to 
have  a  swinging  motion  ;  first,  they  move  forward  or 
toward  the  ea,st  through  a  pretty  long  arc,  then  backward 
or  westward  through  a  short  one,  then  forward  through 
a  longer  one,  etc.  It  is  only  by  the  excess  of  the  longer 
arcs  over  the  shorter  ones  that  the  circuit  of  the  heavens 
is  made. 

The  general  motion  of  the  sun,  moon,  and  planets 
among  the  stars  being  toward  the  east,  the  motion  in  this 
direction  is  called  direct  /  whereas  the  occasional  short 
motions  toward  the  west  are  called  retrograde.  During 
the  periods  between  direct  and  retrograde  motion,  the 
planets  will  for  a  short  time  appear  stationary. 

The  planets  Venus  and  Mercury  are  said  to  be  at  great- 
est elongation  when  at  their  greatest  angular  distance  from 
the  sun.  The  elongation  which  occurs  with  the  planet 
east  of  the  sun,  and  therefore  visible  in  the  western  hori- 
zon after  sunset,  is  called  the  eastern  elongation,  the  other 
the  western  one. 

A  planet  is  said  to  be  in  conjunction  with  the  sun  when 
it  is  in  the  same  direction,  or  when,  as  it  seems  to  pass  by 


ARRANGEMENT  OF  THE  PLANETS. 


115 


the  snn,  it  approaches  nearest  to  it.  It  is  said  to  be  in 
opposition  to  the  sun  when  exactly  in  the  opposite  direc- 
tion— rising  when  the  sun  sets,  and  vice  versa.  If,  when 
a  planet  is  in  conjunction,  it  is  between  the  earth  and  the 
sun,  the  conjunction  is  said  to  be  an  inferior  one  ;  if  be- 
yond the  sun,  it  is  said  to  be  superior. 

Arrangements  and  Motions  of  the  Planets. — We  now 
know  that  the  sun  is  the  real  centre  of  the  solar  system, 
and  that  the  planets  proper  all  revolve  around  it  as  the 
centre  of  motion.  The  order  of  the  five  innermost  large 
planets,  or  the  relative  positions  of  their  orbits,  are  shown 
in  Fig.  48.  These  orbits  are  all  nearly,  but  not  exactly, 


FlG.    48. — ORBITS  OF   THE   PLANETS. 

in  the  same  plane.  The  planets  Mercury  and  Venus 
which,  as  seen  from  the  earth,  never  appear  to  recede  very 
far  from  the  sun,  are  in  reality  those  which  revolve  inside 


116  ASTRONOMY. 

the  orbit  of  the  earth.  The  planets  of  the  third  class, 
which  perform  their  circuits  at  all  distances  from  the  sun, 
are  what  we  now  call  the  superior  planets,  and  are  more 
distant  from  the  sun  than  the  earth  is.  Of  these,  the  or- 
bits of  Mars,  Jupiter,  and  a  swarm  of  telescopic  planets 
are  shown  in  the  figure  ;  next  outside  of  Jupiter  comes 
Saturn,  the  farthest  planet  readily  visible  to  the  naked 
eye,  and  then  Uranus  and  Neptune,  telescopic  planets. 
On  the  scale  of  Fig.  48  the  orbit  of  Neptune  would  be 
more  than  two  feet  in  diameter.  Finally,  the  moon  is  a 
small  planet  revolving  around  the  earth  as  its  centre,  and 
carried  with  the  latter  as  it  moves  around  the  sun. 

Inferior  planets  are  those  whose  orbits  lie  inside  that 
of  the  earth,  as  Mercury  and  Venus. 

Superior  planets  are  those  whose  orbits  lie  outside  that 
of  the  earth,  as  Mars,  Jupiter,  Saturn,  etc. 

The  farther  a  planet  is  situated  from  the  sun,  the  slower 
is  its  orbital  motion.  Therefore,  as  we  go  from  the  sun, 
the  periods  of  revolution  are  longer,  for  the  double  reason 
that  the  planet  has  a  larger  orbit  to  describe  and  moves 
more  slowly  in  its  orbit.  It  is  to  this  slower  motion  of  the 
outer  planets  that  the  occasional  apparent  retrograde  motion 
of  the  planets  is  due,  as  may  be  seen  by  studying  Fig.  49. 
"We  first  remark  that  the  apparent  direction  of  a  planet, 
as  seen  from  the  earth,  is  determined  by  the  line  joining 
the  earth  and  planet.  Supposing  this  line  to  be  continued 
onward  to  infinity,  so  as  to  intersect  the  celestial  sphere, 
the  apparent  motion  of  the  planet  will  be  defined  by  the 
motion  of  the  point  in  which  the  line  intersects  the  sphere. 
If  this  motion  is  toward  the  east,  it  will  be  direct  ;  if 
toward  the  west,  retrograde. 

Let  us  first  take  the  case  of  an  inferior  planet.  Sup- 
pose H 1 K L  M N  to  be  successive  positions  of  the  earth 
in  its  orbit,  and  A  B  C  D  E F  to  be  corresponding  posi- 
tions of  Venus  or  Mercury.  It  must  be  remembered  that 
when  we  speak  of  east  and  west  in  this  connection,  we  do 
not  mean  an  absolute  direction  in  space^  but  a  direction 


APPARENT  MOTIONS  OF  THE  PLANETS. 


11? 


around  the  sphere.  In  the  figure  we  are  supposed  to  look 
down  upon  the  planetary  orbits  from  the  north,  and  a 
direction  west  is,  then,  that  in  which  the  hands  of  a  watch 
move,  while  east  is  in  the  opposite  direction.  When  the 
earth  is  at  H  the  planet  is  seen  at  A .  The  line  H  A 
being  supposed  tangent  to  the  orbit  of  the  planet,  it  is 
evident  from  geometrical  considerations  that  this  is  the 
greatest  angle  which  the  planet  can  ever  make  with  the 
sun  as  seen  from  the  earth.  This,  therefore,  corresponds 
to  the  greatest  eastern  elongation. 


FIG.  49. 

When  the  earth  has  reached  /the  planet  is  at  B,  and  is 
therefore  near  the  direction  IB.  The  line  has  turned  in  a 
direction  opposite  that  of  the  hands  of  a  watch,  and  cuts 
the  celestial  sphere  at  a  point  farther  east  than  the  line 
II A  did.  Hence  the  motion  of  the  planet  during  this 
period  has  been  direct  ;  but  the  direction  of  the  sun  hav- 
ing changed  also  in  consequence  of  the  advance  of  the 
earth,  the  angular  distance  between  the  sun  and  the  planet 
is  less  than  before. 

While  the  earth  is  passing  from  /  to  .7T,  the   planet 


118  ASTRONOMY. 

passes  from  E  to  C.  The  distance  B  C  exceeds  /J^  be- 
cause the  planet  moves  faster  than  the  earth.  The  line 
joining  the  earth  and  planet,  therefore,  cuts  the  celestial 
sphere  at  a  point  farther  west  than  it  did  before,  and 
therefore  the  direction  of  the  apparent  motion  is  retro- 
grade. At  C  the  planet  is  in  inferior  conjunction.  The 
retrograde  motion  still  continues  until  the  earth  reaches  Z, 
and  the  planet  D,  when  it  becomes  stationary.  After- 
ward it  is  direct  until  the  two  bodies  again  corne  into  the 
relative  positions  IB. 

Let  us  next  suppose  that  the  inner  orbit  A  B  CD  EF 
represents  that  of  the  earth,  and  the  outer  one  that  of  a 
superior  planet,  Mars  for  instance.  We  may  consider 
0  Q  P  R  to  be  the  celestial  sphere,  only  it  should  be  infi- 
nitely distant.  While  the  earth  is  moving  from  A  to  B  the 
planet  moves  from  II  to  I.  This  motion  is  direct,  the  di- 
rection 0  Q  P  It  being  from  west  to  east.  While  the  earth 
is  moving  from  B  to  D,  the  planet  is  moving  from  I  to 
L  ;  the  former  motion  being  the  more  rapid,  the  earth 
now  passes  by  the  planet  as  it  were,  and  the  line  conjoin- 
ing them  turns  in  the  same  direction  as  the  hands  of  a 
watch.  Therefore,  during  this  time  the  planet  seems  to 
describe  the  arc  P  Q  in  the  celestial  sphere  in  the  direction 
opposite  to  its  actual  orbital  motion.  The  lines  L  D  and 
M  E  are  supposed  to  be  parallel.  The  planet  is  then  really 
stationary,  even  though  as  drawn  it  would  seem  to  have  a 
forward  motion,  owing  to  the  distance  of  these  two  lines, 
yet,  on  the  infinite  sphere,  this  distance  appears  as  a 
point.  From  the  point  M  the  motion  is  direct  until  the 
two  bodies  once  more  reach  the  relative  positions  B  I. 
When  the  planet  is  at  K  and  the  earth  at  (7,  the  former  is 
in  opposition.  Hence  the  retrograde  motion  of  the  supe- 
rior planets  always  takes  place  near  opposition. 

Theory  of  Epicycles. — The  ancient  astronomers  repre- 
sented this  oscillating  motion  of  the  planets  in  a  way  which 
was  in  a  certain  sense  correct.  The  only  error  they  made 
was,  in  attributing  the  oscillation  to  a  motion  of  the  planet 


APPARENT  MOTIONS  OF  THE  PLANETS.         119 

instead  of  a  motion  of  the  earth  around  the  sun,  which 
really  causes  it.  But  their  theory  was,  notwithstanding, 
the  means  of  leading  COPERNICUS  and  others  to  the  percep- 
tion of  the  true  nature  of  the  motion.  We  allude  to  the 
celebrated  theory  of  epicycles,  by  which  the  planetary 
motions  were  always  represented  before  the  time  of  COPEK- 
NICUS.  Complicated  though  these  motions  were,  it  was 
seen  by  the  ancient  astronomers  that  they  could  be  repre- 
sented by  a  combination  of  two  motions.  First,  a  small 
circle  or  epicycle  was  supposed  to  move  around  the  earth 
with  a  regular,  though  not  uniform,  forward  motion,  and 
then  the  planet  was  supposed  to  move  around  the  circum- 
ference of  this  circle.  The  relation  of  this  theory  to  the 
true  one  was  this.  The  regular  forward  motion  of  the 
epicycle  represents  the  real  motion  of  the  planet  around 
the  sun,  while  the  motion  of  the  planet  around  the  cir- 
cumference of  the  epicycle  is  an  apparent  one  arising 
from  the  revolution  of  the  earth  around  the  sun.  To  ex- 
plain this  we  must  understand  some  of  the  laws  of  relative 
motion. 

It  is  familiarly  known  that  if  an  observer  in  unconscious 
motion  looks  upon  an  object  at  rest,  the  object  will  ap- 
pear to  him  to  move  in  a  direction  opposite  that  in 
which  he  moves.  As  a  result  of  this  law,  if  the  observer 
is  unconsciously  describing  a  circle,  an  object  at  rest  will 
appear  to  him  to  describe  a  circle  of  equal  size.  This  is 
shown  by  the  following  figure.  Let  S  represent  the  sun, 
and  A  B  CD  EF  the  orbit  of  the  earth.  Let  us  suppose 
the  observer  on  the  earth  carried  around  in  this  orbit,  but 
imagining  himself  at  rest  at  8,  the  centre  of  motion. 
Suppose  he  keeps  observing  the  direction  and  distance  of 
the  planet  P,  which  for  the  present  we  suppose  to  be  at 
rest,  since  it  is  only  the  apparent  motion  that  we  shall 
have  to  consider.  "When  the  observer  is  at  A  he  really 
sees  the  planet  in  a  direction  and  distance  A  P,  but 
imagining  himself  at  /S  he  thinks  he  sees  the  planet  at 
the  point  a  determined  by  drawing  a  line  8  a  parallel  and 


120 


ASTRONOMY. 


equal  to  A  P.     As  he  passes  from  A  to  B  tlie  planet 
will  seem  to  him  to  move  in  the  opposite  direction  from 

A  to  J,  the  point  1>  being  deter- 
mined by  drawing  Sb  equal  and 
parallel  to  B  P.  As  he  recedes 
from  the  planet  through  the  arc 
BCD,  the  planet  seems  to  re- 
cede from  him  through  l>  c  d  ; 
and  while  he  moves  from  left  to 
right  through  DE  the  planet 
seems  to  move  from  right  to  left 
through  D  E.  Finally,  as  he  ap- 
proaches the  planet  through  the 
arc  E  F A  the  planet  seems  to 
approach  him  through  EFA, 
and  when  he  returns  to  A  the 
planet  will  appear  at  A9  as  in  the 
beginning.  Thus  the  planet, 
though  really  at  rest,  will  seem 
to  him  to  move  over  the  circle 
abcdef  corresponding  to  that 
in  which  the  observer  himself  is 
carried  around  the  sun. 

The  planet  being  really  in  motion,  it  is  evident  that 
the  combined  effect  of  the  real  motion  of  the  planet  and 
the  apparent  motion  around  the  circle  a  b  c  d  ef  will  be 
represented  by  carrying  the  centre  of  this  circle  P  along 
the  true  orbit  of  the  planet.  The  motion  of  the  earth 
being  more  rapid  than  that  of  an  outer  planet,  it  follows 
that  the  apparent  motion  of  the  planet  through  a  b  is  more 
rapid  than  the  real  motion  of  P  along  the  orbit.  Hence 
in  this  part  of  the  orbit  the  movement  of  the  planet  will  be 
retrograde.  In  every  other  part  it  will  be  direct,  because 
the  progressive  motion  of  P  will  at  least  overcome,  some- 
times be  added  to,  the  apparent  motion  around  the  circle. 
In  the  ancient  astronomy  the  apparent  small  circle 
abcdef  was  called  the  epicycle. 


FIG.  50. 


UNEQUAL  MOTION  OF  THE  PLANETS.  121 

In  the  case  of  the  inner  planets  Mercury  and  Venus 
the  relation  of  the  epicycle  to  the  true  orbit  is  reversed. 
Here  the  epicyclic  motion  is  that  of  the  planet  around 
its  real  orbit — that  is,  the  true  orbit  of  the  planet  around 
the  sun  was  itself  taken  for  the  epicycle,  while  the 
forward  motion  was  really  due  to  the  apparent  revolu- 
tion of  the  sun  produced  by  the  annual  motion  of  the 
earth. 

In  the  preceding  descriptions  of  the  planetary  motions 
we  have  spoken  of  them  all  as  circular.  But  it  was  found 
by  HIPP ARCHUS  *  that  none  of  the  planetary  motions  were 
really  uniform.  Studying  the  motion  of  the  sun  in  order 
to  determine  the  length  of  the  year,  he  observed  the  times 
of  its  passage  through  the  equinoxes  and  solstices  with  all 
the  accuracy  which  his  instruments  permitted.  He  found 
that  it  was  several  days  longer  in  passing  through  one  half 
of  its  course  than  through  the  other.  This  was  apparently 
incompatible  with  the  favorite  theory  of  the  ancients  that 
all  the  celestial  motions  were  circular  and  uniform.  It 
was,  however,  accounted  for  by  supposing  that  the  earth 
was  not  in  the  centre  of  the  circle  around  which  the  sun 
moved,  but  a  little  to  one  side.  Thus  arose  the  cele- 
brated theory  of  the  eccentric.  Careful  observations  of 
the  planets  showed  that  they  also  had  similar  inequalities 
of  motion.  The  centre  of  the  epicycle  around  which  the 
real  planet  was  carried  was  found  to  move  more  rapidly 
in  one  part  of  the  orbit,  and  more  slowly  in  the  opposite 
part.  Thus  the  circles  in  which  the  planets  were  sup- 
posed to  move  were  not  truly  centred  upon  the  earth. 
They  were  therefore  called  eccentrics. 

This  theory  accounted  in  a  rough  way  for  the  observed 
inequalities.  It  is  evident  that  if  the  earth  was  supposed 
to  be  displaced  toward  one  side  of  the  orbit  of  the  planet, 

*  HIPPARCHUS  was  one  of  the  most  celebrated  astronomers  of  anti- 
quity, being  frequently  spoken  of  as  the  father  of  the  science.  He  is 
supposed  to  have  made  most  of  his  observations  at  Rhodes,  and  flour- 
ished about  one  hundred  and  fifty  years  before  the  Christian  era. 


122  ASTRONOMY. 

the  latter  would  seem  to  move  more  rapidly  when  nearest 
the  earth  than  when  farther  from  it.  It  was  not  until  the 
time  of  KEPLER  that  the  eccentric  was  shown  to  be  in- 
capable of  accounting  for  the  real  motion  ;  and  it  is  his 
discoveries  which  we  are  next  to  describe. 

§   2.    KEPLER'S  LAWS  OF  PLANETARY  MOTION. 

The  direction  of  the  sun,  or  its  longitude,  can  be  deter- 
mined from  day  to  day  by  direct  observation.  If  we 
could  also  observe  its  distance  on  each  day,  we  should,  by 
laying  down  the  distances  and  directions  on  a  large  piece 
of  paper,  through  a  whole  year,  be  able  to  trace  the  curve 
which  the  earth  describes  in  its  annual  course,  this  course 
being,  as  already  shown,  the  counterpart  of  the  apparent 
one  of  the  sun.  A  rough  determination  of  the  rela- 
tive distances  of  the  sun  at  different  times  of  the  year  may 
be  made  by  measuring  the  sun's  apparent  angular  diame- 
ter, because  this  diameter  varies  inversely  as  the  distance 
of  the  object  observed.  Such  measures  would  show  that 
the  diameter  was  at  a  maximum  of  32'  36"  on  January  1st, 
and  at  a  minimum  of  31/  32"  on  July  1st  of  every  year. 
The  difference,  64",  is,  in  round  numbers,  -^  the  mean 
diameter — that  is,  the  earth  is  nearer  the  sun  on  January 
1st  than  on  July  1st  by  about  -fa .  We  may  consider  it 
as  -^  greater  than  the  mean  on  the  one  date,  and  -fa  less 
on  the  other.  This  is  therefore  the  actual  displacement 
of  the  sun  from  the  centre  of  the  earth's  orbit. 

Again,  observations  of  the  apparent  daily  motion  of 
the  sun  among  the  stars,  corresponding  to  the  real  daily 
motion  of  the  earth  round  the  sun,  show  this  motion  to  be 
least  about  July  1st,  when  it  amounts  to  57'  12"  =  3432", 
and  greatest  about  January  1st,  when  it  amounts  to 
61'  11"  =  3671".  The  difference,  239",  is,  in  round  num- 
bers, ^  the  mean  motion,  so  that  the  range  of  variation 
is,  in  proportion  to  the  mean,  double  what  it  is  in  the  case 
of  the  distances.  If  the  actual  velocity  of  the  earth  in  its 


KEPLER'S  LAWS.  123 

orbit  were  uniform,  the  apparent  angular  motion  round 
the  sun  would  be  inversely  as  its  distance  from  the  sun. 
Actually,  however,  the  angular  motion,  as  given  above,  is 
inversely  as  the  square  of  the  distance  from  the  sun,  be- 
cause (1  -|-  -gV)2  —  1  +  IT  verJ  nearly.  The  actual  ve- 
locity of  the  earth  is  therefore  greater  the  nearer  it  is  to 
the  sun. 

On  the  ancient  theory  of  the  eccentric  circle,  as  pro- 
pounded by  HIPPARCHUS,  the  actual  motion  of  the  earth 
was  supposed  to  be  uniform,  and  it  was  necessary  to  sup- 
pose the  displacement  of  the  sun  (or,  on  the  ancient  theo- 
ry, of  the  earth)  from  the  centre  to  be  -fa  its  mean  distance, 
in  order  to  account  for  the  observed  changes  in  the  motion 
in  longitude.  We  now  know  that,  in  round  numbers,  one 
half  the  inequality  of  the  apparent  motion  of  the  sun  in 
longitude  arises  from  the  variations  in  the  distance  of  the 
earth  from  it,  and  one  half  from  the  earth's  actually  mov- 
ing with  a  greater  velocity  as  it  comes  nearer  the  sun.  By 
attributing  the  whole  inequality  to  a  variation  of  distance, 
the  ancient  astronomers  made  the  eccentricity  of  the  or- 
bit— that  is,  the  distance  of  the  sun  from  the  geometrical 
centre  of  the  orbit  (or,  as  they  supposed,  the  distance  of 
the  earth  from  the  centre  of  the  sun's  orbit) — twice  as 
great  as  it  really  was. 

An  immediate  consequence  of  these  facts  of  observa- 
tion is  KEPLER'S  second  law  of  planetary  motion,  that  the 
radii  vectores  drawn  from  the  sun  to  a  planet  revolving 
round  it,  sweep  over  equal  areas  in  equal  times.  Sup- 
pose, in  Fig.  51,  that  S  represents  the  position  of  the  sun, 
and  that  the  earth,  or  a  planet,  in  a  unit  of  time,  say  a 
day  or  a  week,  moves  from  jP2  to  Pz.  At  another  part 
of  its  orbit  it  moves  from  P  to  Pl  in  the  same  time, 
and  at  a  third  part  from  jP4  to  P6.  Then  the  areas 
SP,P,,  SPP»  SP>P>  will  all  be  equal.  A  little 
geometrical  consideration  will,  in  fact,  make  it  clear  that 
the  areas  of  the  triangles  are  equal  when  the  angles  at  S 
are  inversely  as  the  square  of  the  radii  vectores,  S P,  etc., 


124  ASTRONOMY. 

since  the  expression  for  the  area  of  a  triangle  in  which  the 
angle  at  S  is  very  small  is  -J  angle  8  X 


FlG.    61. — LAW   OP   AREAS. 

In  the  time  of  KEPLER  the  means  of  measuring  the 
sun's  angular  diameter  were  so  imperfect  that  the  preced- 
ing method  of  determining  the  path  of  the  earth  around 
the  sun  could  not  be  applied.  It  was  by  a  study  of  the 
motions  of  the  planet  Mars,  as  observed  by  TYCHO  BRAHE, 
that  KEPLER  was  led  to  his  celebrated  laws  of  planetary 
motion.  He  found  that  no  possible  motion  of  Mars  in  a 
truly  circular  orbit,  however  eccentric,  would  represent  the 
observations.  After  long  and  laborious  calculations,  and 
the  trial  and  rejection  of  a  great  number  of  hypotheses, 
he  was  led  to  the  conclusion  that  the  planet  Mars  moved 
in  an  ellipse,  having  the  sun  in  one  focus.  As  the  analo- 
gies of  nature  led  to  the  inference  that  all  the  planets, 
the  earth  included,  moved  in  curves  of  the  same  class, 
and  according  to  the  same  law,  he  was  led  to  enunciate 
the  first  two  of  his  celebrated  laws  of  planetary  motion, 
which  were  as  follow  : 

*  More  exactly  if  we  consider  the  arc  PPi  as  a  straight  line,  the 
area  of  the  triangle  PPl  S  will  be  equal  to  1 8Px  8 Pi  x  sin  angle  S. 
But  in  considering  only  very  small  angles  we  may  suppose  8P=  SPi 
and  the  sine  of  the  angle  8  equal  to  the  angle  itself.  This  supposition 
will  give  the  area  mentioned  above. 


KEPLER'S  LAWS.  125 

I.  Each  planet  moves  around  the  sun  in  an  ellipse,  hav- 
ing the  sun  in  one  of  its  foci. 

II.  The    radius  vector  joining  each  planet  with  the 
sun,  moves  over  equal  areas  in  equal  times. 

To  these  be  afterward  added  another  showing  the  rela- 
tion between  the  times  of  revolution  of  the  separate 
planets. 

III.  The    square   of  the   time   of  revolution   of  each 
planet  is  proportional  to  the  cube  of  its  mean  distance 
from  the  sun. 

These  three  laws  comprise  a  complete  theory  of  plan- 
etary motion,  so  far  as  the  main  features  of  the  motion  are 
concerned.  There  are,  indeed,  small  variations  from 
these  laws  of  KEPLER,  but  the  laws  are  so  nearly  correct 
that  they  are  always  employed  by  astronomers  as  the  basis 
of  their  theories. 

Mathematical  Theory  of  the  Elliptic  Motion. —  The 
laws  of  KEPLER  lead  to  problems  of  such  mathematical 
elegance  that  we  give  a  brief  synopsis  of  the  most  impor- 
tant elements  of  the  theory.  A  knowledge  of  the  ele- 
ments of  analytic  geometry  is  necessary  to  understand  it. 

Let  us  put  : 

a,  the  semi-major  axis  of  the  ellipse  in  which  the  planet  moves. 
In  the  figure,  if  0  is  the  centre  of  the  el- 
lipse, and  S  the  focus  in  which  the  sun  is 
situated,  then  a  =  A  C  =  C  n. 

f1  **! 

e,  the  eccentricity  of  the  ellipse  =  — -. 

TT,  the  longitude  of  the  perihelion,  rep- 
resented by  the  angle  K  S  E,  E  being  the 
direction  of  the  vernal  equinox  from 
which  longitudes  are  counted. 

7i,  the  mean  angular  motion  of  the 
planet  round  the  sun  in  a  unit  of  time. 
The  actual  motion  being  variable,  the 
mean  motion  is  found  by  dividing  the  FIG.  52. 

circumference  =  360°  by  the  time  of  revolution. 

T1  the  time  of  revolution. 

r,  the  distance  of  the  planet  from  the  sun,  or  its  radius  vector,  a 
variable  quantity. 

I.  The  first  remark  we  have  to  make  is  that  the  ellipticities  of  the 


126  ASTRONOMY. 

planetary  orbits  —  that  is,  the  proportions  in  which  the  orbits  are  flat- 
tened —  is  much  less  than  their  eccentricities.  By  the  properties  of 
the  ellipse  we  have  : 

SB  =  semi-major  axis  =  a, 
BC=  semi-minor  axis  =  a  V\  —  e*, 
or,  B  0  =  a  (1  —  \  e2)  nearly,  when  e  is  very  small. 

The  most  eccentric  of  the  orbits  of  the  eight  major  planets  is  that 
of  Mercury,  for  which  e  =  0.2.     Hence  for  Mercury 


very  nearly,  so  that  flattening  of  the  orbit  is  only  about  -^  or  .02 
of  the  major  axis. 

The  next  most  eccentric  orbit  is  that  of  Mars  for  which  e  =  .093  ; 
B  C  =  a  (1  —  .0043),  so  that  the  flattening  of  the  orbit  is  only 
about  2^-5. 

We  see  from  this  that  the  hypothesis  of  the  eccentric  circle  makes 
a  very  close  approximation  to  the  true  form  of  the  planetary  orbits. 
It  is  only  necessary  to  suppose  the  sun  removed  from  the  centre  of 
the  orbit  by  a  quantity  equal  to  the  product  of  the  eccentricity  into 
the  radius  of  the  orbit  to  have  a  nearly  true  representation  of  the 
orbit  and  of  the  position  of  the  sun. 

II.  The  least  distance  of  the  planet  from  the  sun  is 

Sn  =  «(!-«), 
and  the  greatest  distance  is 

A  S  =  a  (I  +  e). 

III.  The  angular  velocity  of  the  planet  around  the  sun  at  any 
point  of  the  orbit,  which  we  may  call  8,  is,  by  the  second  law  of 
KEPLER  : 

-S- 

C  being  a  constant  to  be  determined.  To  determine  it  we  remark 
that  8  is  the  angle  through  which  the  planet  moves  in  a  unit  of 
time.  If  we  suppose  this  unit  to  be  very  small,  the  quantity  8  r*  is 
double  the  area  of  the  very  small  triangle  swept  over  by  the  radius 
vector  during  such  unit.  This  area  is  called  the  areolar  velocity  of 
the  planet,  and  is  a  constant,  by  KEPLER'S  second  law.  Therefore, 
in  the  last  equation,  G  —  S  r2  represents  the  double  of  the  areolar 
velocity  of  the  planet.  When  the  planet  completes  an  entire  revo- 
lution, the  radius  vector  has  swept  over  the  whole  area  of  the 
ellipse  which  is  ?r  «2  V  1  —  e2.*  The  time  required  to  do  this  be- 

*  In  this  formula  ?r  represents  the  ratio  of  the  circumference  of  the 
circle  to  its  diameter. 


KEPLER' 8  LAWS.  127 

ing  called  T7,  the  area  swept  over  with  the  areolar  velocity  \  C  is 
also  \G  T.     Therefore 


_ 

(J  = 


The  quantity  2  TT  here  represents  360°,  or  the  whole  circumference, 
which,  being  divided  by  T,  the  time  of  describing  it  will  give  the 
mean  angular  velocity  of  the  planet  around  the  sun  which  we  have 
called  n.  Therefore 

_  2jr 

and 

C  =  a?n  VT^7». 

This  value  of  C  being  substituted  in  the  expression  for  8,  we  have 

««^*S 


IV.    By  KEPLER'S  third  law  T*  is  proportioned  to  «3  ;  that  is, 

7*2 
—  —  is  a  constant  for  all  the  planets.     The  numerical  value  of  this 

a* 

constant  will  depend  upon  the  quantities  which  we  adopt  as  the  units 
of  time  and  distance.     If  we  take  the  year  as  the  unit  of  time  and 

the  mean  distance  of  the  earth  from  the  sun  as  that  of  distance,  T 

jn 

and  a  for  the  earth  will  both  be  unity,  and  the  ratio  —5-  will    there- 
fore be  unity  for  all  the  planets.     Therefore 


Therefore  if  we  square  the  period  of  revolution  of  any  planet  in  yearn, 
and  extract  the  cube  root  of  the  square,  we  shall  have  its  mean  distance 
from  the  sun  in  units  of  the  earth's  distance. 

It  is  thus  that  the  mean  distances  of  the  planets  are  determined 
in  practice,  because,  by  a  long  series  of  observations,  the  times  of 
revolution  of  the  planets  have  been  determined  with  very  great  pre- 
cision. 

V.  To  find  the  position  of  a  planet  we  must  know  the  epoch  at 
which  it  passed  its  perihelion,  or  some  equivalent  quantity.  To 
find  its  position  at  any  other  time  let  T  be  the  time  which  has  elapsed 
since  passing  the  perihelion.  Then,  by  the  law  of  areas,  if  P  be  the 
position  of  the  planet  at  this  time  we  shall  have 

Area  of  sector  P  Sir         T 
Area  of  whole  ellipse  ~~  T 


128  ASTRONOMY. 

The  times  r  and  T  being  both  given,  the  problem  is  reduced  to 
that  of  cutting  a  given  area  of  the  ellipse  by  a  line  drawn  from  the 
focus  to  some  point  of  its  circumference  to  be  found.  This  is 
known  as  KEPLER'S  problem,  and  may  be  solved  by  analytic  geoni- 


FiG.  53. 

etry  as  follows  :  Let  A  B  be  the  major  axis  of  the  ellipse,  Pthe 
position  of  the  planet,  and  S  that  of  the  focus  in  which  the  sun  is 
situated.  On  A  B  as  a  diameter  describe  a  circle,  and  through  P 
draw  the  right  line  P*  P  D  perpendicular  to  A  B. 

The  area  of  the  elliptic  sector  SPB,  over  which  the  radius  vector 
of  the  planet  has  swept  since  the  planet  passed  the  perihelion  at  B, 
is  equal  to  the  sector  G P  B  minus  the  triangle  GPS.  Since  an 
ellipse  is  formed  from  a  circle  by  shortening  all  the  ordinates  in 
the  same  ratio  (namely,  the  ratio  of  the  minor  axis  It  to  the  major 
axis  «),  it  follows  that  the  elliptic  sector  0  P  B  may  be  formed 
from  the  circular  sector  C  P*  B  by  shortening  all  the  ordinates  in 
the  ratio  of  I)  P  to  D  .P,  or  of  a  to  &.  Hence, 

Area  CPB  :  area  C  P'  B  =  b  :  a. 

But  area  C  P*  B  =  angle  P/  C  B  x  i  a2,  taking  the  unit  radius 
as  the  unit  of  angular  measure.  Hence,  putting  u  for  the  angle 
P*  C  B  we  have 

Area  CPB  =  -  area  C  P'  B  =  \  a  I  u  (2). 

a 

Again,  the  area  of  the  triangle  CP  S  is  equal  to  £  base  C  S   x    al- 
titude P  D.     Also  P  D  =  -  P/  D,  and  P'  D  =  C  P*  sin  u  —  a  sin  u. 
a 

Wherefore, 

1tsmu  (3). 


KEPLER'S  LAWS.  129 

By  the  first  principles  of  conic  sections,  C  8,  the  base  of  the 
triangle,  is  equal  to  a  e.  Hence 

Area  GPS  =  %abtsinu, 
and,  from  (2)  and  (3), 

Area  SPJB  =  ^ab(u  —  esin  w). 

Substituting  in  equation  (1)  this  value  of  the  sector  area,  and 
TT  a  b  for  the  area  of  the  ellipse,  we  have 

u  —  e  sin  u       r 

2?r          ~  T" 
or, 

u  —  e  sin  u  =  2  TT  — . 

From  this  equation  the  unknown  angle  u  is  to  be  found.  The 
equation  being  a  transcendental  one,  this  cannot  be  done  directly, 
but  it  may  be  rapidly  done  by  successive  approximation,  or  the 
value  of  u  may  be  developed  in  an  infinite  series. 

Next  we  wish  to  express  the  position  of  the  planet,  which  is  given 
by  its  radius  vector  8  P  and  the  angle  B  8  P  which  this  radius 
vector  makes  with  the  major  axis  of  the  orbit.  Let  us  put 

r,  the  radius  vector  SP, 

/,  the  angle  B  SP,  called  the  true  anomaly. 

Then 

r  sin/  =  PD  =  b  sin  u  (Equation  3), 

rcoaf=SD=  CD  —  03=  0 P'  cos  u  —  ae  =  a  (cosu  —  e), 

from  which  r  and  /  can  both  be  determined.  By  taking  the  square 
root  of  the  sums  of  the  squares,  they  give,  by  suitable  reduction  and 
putting  &a  =  d"  (1  —  02), 

r  —  a  (1  —  e  cos  w), 
and,  by  dividing  the  first  by  the  second, 

b  sin  u  Vl  —  e"*  sin  u 


tftn/±= 


a  (cos  u  —  e)  cos  u  —  e 


Putting,  as  before,  TT  for  the  longitude  of  the  perihelion,  the  true 
longitude  of  the  planet  in  its  orbit  will  be/  +  TT. 

VI.  To  find  the  position  of  the  planet  relatively  to  the  ecliptic, 


130  ASTRONOMY. 

the  inclination  of  the  orbit  to  the  ecliptic  has  to  be  taken  into  ac- 
count. The  orbits  of  the  several  large  planets  do  not  lie  in  the 
same  plane,  but  are  inclined  to  each  other,  and  to  the  ecliptic,  by 
various  small  angles.  A  table  giving  the  values  of  these  angles 
will  be  given  hereafter,  from  which  it  will  be  seen  that  the  orbit  of 
Mercury  has  the  greatest  inclination,  amounting  to  7°,  and  that  of 
Uranus  the  least,  being  only  46'.  The  reduction  of  the  position  of 
the  planet  to  the  ecliptic  is  a  problem  of  spherical  trigonometry, 
the  solution  of  which  need  not  be  discussed  here. 


CHAPTER    V. 

UNIVERSAL   GRAVITATION. 
§  1.    NEWTON'S  LAWS  OP  MOTION. 

THE  establishment  of  the  theory  of  universal  gravitation 
furnishes  one  of  the  best  examples  of  scientific  method 
which  is  to  be  found.  We  shall  describe  its  leading 
features,  less  for  the  purpose  of  making  known  to  the 
reader  the  technical  nature  of  the  process  than  for  illus- 
trating the  true  theory  of  scientific  investigation,  and 
showing  that  such  investigation  has  for  its  object  the  dis- 
covery of  what  we  may  call  generalized  facts.  The  real 
test  of  progress  is  found  in  our  constantly  increased 
ability  to  foresee  either  the  course  of  nature  or  the  effects 
of  any  accidental  or  artificial  combination  of  causes.  So 
long  as  prediction  is  not  possible,  the  desires  of  the  inves- 
tigator remain  unsatisfied.  When  certainty  of  prediction 
is  once  attained,  and  the  laws  on  which  the  prediction  is 
founded  are  stated  in  their  simplest  form,  the  work  of 
science  is  complete. 

The  whole  process  of  scientific  generalization  consists  in 
grouping  facts,  new  and  old,  under  such  general  laws  that 
they  are  seen  to  be  the  result  of  those  laws,  combined  with 
those  relations  in  space  and  time  which  we  may  suppose  to 
exist  among  the  material  objects  investigated.  It  is  essen- 
tial to  such  generalization  that  a  single  law  shall  suffice  for 
grouping  and  predicting  several  distinct  facts.  A  law 
invented  simply  to  account  for  an  isolated  fact,  however 


132  ASTRONOMY. 

general,  cannot  be  regarded  in  science  as  a  law  of  nature. 
It  may,  indeed,  be  true,  but  its  truth  cannot  be  proved 
until  it  is  shown  that  several  distinct  facts  can  be  accounted 
for  by  it  better  than  by  any  other  law.  The  reader  will 
call  to  mind  the  old  fable  which  represented  the  earth  as 
supported  on  the  back  of  a  tortoise,  but  totally  forgot  that 
the  support  of  the  tortoise  needed  to  be  accounted  for  as 
much  as  that  of  the  earth. 

To  the  pre-Newtonian  astronomers,  the  phenomena  of  the 
geometrical  laws  of  planetary  motion,  which  we  have  just 
described,  formed  a  group  of  facts  having  no  connection 
with  any  thing  on  the  earth.  The  epicycles  of  HIPPAECHUS 
and  PTOLEMY  were  a  truly  scientific  conception,  in  that  they 
explained  the  seemingly  erratic  motions  of  the  planets  by 
a  single  simple  law.  In  the  heliocentric  theory  of  COPER- 
NICUS this  law  was  still  further  simplified  by  dispensing  in 
great  part  with  the  epicycle,  and  replacing  the  latter  by  a 
motion  of  the  earth  around  the  sun,  of  the  same  nature 
with  the  motions  of  the  planets.  But  COPERNICUS  had  no 
way  of  accounting  for,  or  even  of  describing  with  rigor- 
ous accuracy,  the  small  deviations  in  the  motions  of  the 
planets  around  the  sun.  In  this  respect  he  made  no  real 
advance  upon  the  ideas  of  the  ancients. 

KEPLER,  in  his  discoveries,  made  a  great  advance 
in  representing  the  motions  of  all  the  planets  by  a 
single  set  of  simple  and  easily  understood  geometrical 
laws.  Had  the  planets  followed  his  laws  exactly,  the 
theory  of  planetary  motion  would  have  been  substantially 
complete.  Still,  further  progress  was  desired  for  two 
reasons.  In  the  first  place,  the  laws  of  KEPLER  did  not 
perfectly  represent  all  the  planetary  motions.  When  ob- 
servations of  the  greatest  accuracy  were  made,  it  was  found 
that  the  planets  deviated  by  small  amounts  from  the  ellipse 
of  KEPLER.  Some  small  emendations  to  the  motions  com- 
puted on  the  elliptic  theory  were  therefore  necessary. 
Had  this  requirement  been  fulfilled,  still  another  step 
would  have  been  desirable — namely,  that  of  connecting  the 


LA  WS  OF  MOTION.  133 

motions  of  the  planets  with  motion  upon  the  earth,  and 
reducing  them  to  the  same  laws. 

Notwithstanding  the  great  step  which  KEPLER  made  in 
describing  the  celestial  motions,  he  unveiled  none  of  the 
great  mystery  in  which  they  were  enshrouded.  This  mys- 
tery was  then,  to  all  appearance,  impenetrable,  because 
not  the  slightest  likeness  could  be  perceived  between  the 
celestial  motions  and  motions  on  the  surface  of  the  earth. 
The  difficulty  was  recognized  by  the  older  philosophers  in 
the  division  of  motions  into  "  forced  "  and  "natural." 
The  latter,  they  conceived,  went  on  perpetually  from  the 
very  nature  of  things,  while  the  former  always  tended  to 
cease.  So  when  KEPLER  said  that  observation  showed  the 
law  of  planetary  motion  to  be  that  around  the  circum- 
ference of  an  ellipse,  as  asserted  in  his  law,  he  said  all  that 
it  seemed  possible  to  learn,  supposing  the  statement  per- 
fectly exact.  And  it  was  all  that  could  be  learned  from  the 
mere  study  of  the  planetary  motions.  In  order  to  connect 
these  motions  with  those  on  the  earth,  the  next  step  was  to 
study  the  laws  of  force  and  motion  here  around  us.  Sin- 
gular though  it  may  appear,  the  ideas  of  the  ancients  on 
this  subject  were  far  more  erroneous  than  their  concep- 
tions of  the  motions  of  the  planets.  "We  might  almost  say 
that  before  the  time  of  GALILEO  scarcely  a  single  correct 
idea  of  the  laws  of  motion  was  generally  entertained  by 
men  of  learning.  There  were,  indeed,  one  or  two  who  in 
this  respect  were  far  ahead  of  their  age.  LEONARDO  DA 
YINCI,  the  celebrated  painter,  was  noted  in  this  respect. 
But  the  correct  ideas  entertained  by  him  did  not  seem  to 
make  any  headway  in  the  world  until  the  early  part  of 
the  seventeenth  century.  Among  those  who,  before  the 
time  of  NEWTON,  prepared  the  way  for  the  theory  in 
question,  GALILEO,  HUYGHENS,  and  HOOKE  are  entitled  to 
especial  mention.  As,  however,  we  cannot  develop  the 
history  of  this  subject,  we  must  pass  at  once  to  the  gen- 
eral laws  of  motion  laid  down  by  NEWTON.  These  were 
three  in  number. 


134  ASTRONOMY. 

Law  First :  Every  ~body  preserves  its  state  of  rest  or  of 
uniform  'motion  in  a  right  line,  unless  it  is  compelled  to 
change  that  state  by  forces  impressed  thereon. 

It  was  formerly  supposed  that  a  body  acted  on  by  no 
force  tended  to  come  to  rest.  Here  lay  one  of  the  great- 
est difficulties  which  the  predecessors  of  NEWTON  found, 
in  accounting  for  the  motion  of  the  planets.  The  idea 
that  the  sun  in  some  way  caused  these  motions  was  enter- 
tained from  the  earliest  times.  Even  PTOLEMF  had  a 
vague  idea  of  a  force  which  was  always  directed  toward 
the  centre  of  the  earth,  or,  which  was  to  him  the  same 
thing,  toward  the  centre  of  the  universe,  and  which  not 
only  caused  heavy  bodies  to  fall,  but  bound  the  whole  uni- 
verse together.  KEPLEK,  again,  distinctly  affirms  the  ex- 
istence of  a  gravitating  force  by  which  the  sun  acts  on  the 
planets  ;  but  he  supposed  that  the  sun  must  also  exercise 
an  impulsive  forward  force  to  keep  the  planets  in  motion. 
The  reason  of  this  incorrect  idea  was,  of  course,  that  all 
bodies  in  motion  on  the  surface  of  the  earth  had  practically 
come  to  rest.  But  what  was  not  clearly  seen  before  the 
time  of  NEWTON,  or  at  least  before  GALILEO,  was,  that  this 
arose  from  the  inevitable  resisting  forces  which  act  upon 
all  moving  bodies  around  us. 

Law  Second  :  The  alteration  of  motion  is  ever  propor- 
tional to  the  moving  force  impressed,  and  is  made  in  the 
direction  of  the  right  line  in  which  that  force  acts. 

The  first  law  might  be  considered  as  a  particular  case  of 
this  second  one  arising  when  the  force  is  supposed  to  van- 
ish. The  accuracy  of  both  laws  can  be  proved  only  by 
very  carefully  conducted  experiments.  They  are  now 
considered  as  mathematically  proved. 

Law  Third  :  To  every  action  there  is  always  opposed  an 
equal  reaction  •  or  the  mutual  actions  of  two  'bodies  upon 
each  other  are  always  equal,  and  in  opposite  directions. 

That  is,  if  a  body  A  acts  in  any  way  upon  a  body  B, 
B  will  exert  a  force  exactly  equal  on  A  in  the  opposite 
direction. 


GRAVITATION  OF  THE  PLANETS.  135 

These  laws  once  established,  it  became  possible  to  calcu- 
late the  motion  of  any  body  or  system  of  bodies  when  once 
the  forces  which  act  on  them  were  known,  and,  vice  versa, 
to  define  what  forces  were  requisite  to  produce  any  given 
motion.  The  question  which  presented  itself  to  the  mind 
of  NEWTON  and  his  contemporaries  was  this  :  Under  what 
law  of  force  will  planets  move  round  the  sun  in  accord- 
ance with  KEPLER'S  laws  f 

The  laws  of  central  forces  had  been  discovered  by  HUY- 
GHENS  some  time  before  NEWTON  commenced  his  re- 
searches, and  there  was  one  result  of  them  which,  taken  in 
connection  with  KEPLER'S  third  law  of  motion,  was  so 
obvious  that  no  mathematician  could  have  had  much  diffi- 
culty in  perceiving  it.  Supposing  a  body  to  move  around 
in  a  circle,  and  putting  12  the  radius  of  the  circle,  T  the 
period  of  revolution,  HUYGHENS  showed  that  the  centrifugal 
force  of  the  body,  or,  which  is  the  same  thing,  the  attract- 
ive force  toward  the  centre  which  would  keep  it  in  the 

7~> 

circle,  was  proportional  to  ^.  But  by  KEPLER'S  third 
law  T*  is  proportional  to  12\  Therefore  this  centripetal 

D  1 

force  is  proportional  to  -^5,  that  is,  to  -^.  Thus  it  fol- 
lowed immediately  from  KEPLER'S  third  law,  that  the 
central  force  which  would  keep  the  planets  in  their  or- 
bits was  inversely  as  the  square  of  the  distance  from  the 
sun,  supposing  each  orbit  to  be  circular.  The  first  law  of 
motion  once  completely  understood,  it  was  evident  that 
the  planet  needed  no  force  impelling  it  forward  to  keep 
up  its  motion,  but  that,  once  started,  it  would  keep  on 
forever. 

The  next  step  was  to  solve  the  problem,  what  law  of 
force  will  make  a  planet  describe  an  ellipse  around  the 
sun,  having  the  latter  in  one  of  its  foci  ?  Or,  supposing 
a  planet  to  move  round  the  sun,  the  latter  attracting  it 
with  a  force  inversely  as  the  square  of  the  distance  ;  what 
will  be  the  form  of  the  orbit  of  the  planet  if  it  is  not  cir- 


136  ASTRONOMY. 

cular  ?  A  solution  of  either  of  these  problems  was  beyond 
the  power  of  mathematicians  before  the  time  of  NEWTON  ; 
and  it  thus  remained  uncertain  whether  the  planets  mov- 
ing under  the  influence  of  the  sun's  gravitation  would  or 
would  not  describe  ellipses.  Unable,  at  first,  to  reach  a 
satisfactory  solution,  NEWTON  attacked  the  problem  in 
another  direction,  starting  from  the  gravitation,  not  of 
the  sun,  but  of  the  earth,  as  explained  in  the  following 
section. 


§  2.    GRAVITATION  IN  THE  HEAVENS. 

The  reader  is  probably  familiar  with  the  story  of  NEW- 
TON and  the  falling  apple.  Although  it  has  no  authorita- 
tive foundation,  it  is  strikingly  illustrative  of  the  method 
by  which  NEWTON  first  reached  a  solution  of  the  problem. 
The  course  of  reasoning  by  which  he  ascended  from  grav- 
itation on  the  earth  to  the  celestial  motions  was  as  follows  : 
We  see  that  there  is  a  force  acting  all  over  the  earth  by 
which  all  bodies  are  drawn  toward  its  centre.  This  force 
is  familiar  to  every  one  from  his  infancy,  and  is  properly 
called  gravitation.  It  extends  without  sensible  diminution 
to  the  tops  not  only  of  the  highest  buildings,  but  of  the 
highest  mountains.  How  much  higher  does  it  extend  ? 
Why  should  it  not  extend  to  the  moon  ?  If  it  does,  the 
moon  would  tend  to  drop  toward  the  earth,  just  as  a  stone 
thrown  from  the  hand  drops.  As  the  moon  moves  round 
the  earth  in  her  monthly  course,  there  must  be  some  force 
drawing  her  toward  the  earth  ;  else,  by  the  first  law  of 
motion,  she  would  fly  entirely  away  in  a  straight  line.  Why 
should  not  the  force  which  makes  the  apple  fall  be  the 
same  force  which  keeps  her  in  her  orbit  ?  To  answer  this 
question,  it  was  not  only  necessary  to  calculate  the  intensity 
of  the  force  which  would  keep  the  moon  herself  in  her 
orbit,  but  to  compare  it  with  the  intensity  of  gravity  at  the 
earth's  surface.  It  had  long  been  known  that  the  distance 
of  the  moon  was  about  sixty  radii  of  the  earth.  If  this 


GRAVITATION  OF  THE  PLANETS.  137 

force  diminished  as  the  inverse  square  of  the  distance, 
then,  at  the  moon,  it  would  be  only  ^Vir  as  great  as  at 
the  surface  of  the  earth.  On  the  earth  a  body  falls  six- 
teen feet  in  a  second.  If,  then,  the  theory  of  gravitation 
were  correct,  the  moon  ought  to  fall  toward  the  earth 
-g-gLg-  of  this  amount,  or  about  -^  of  an  inch  in  a  second. 
The  moon  being  in  motion,  if  we  imagine  it  moving  in  a 
straight  line  at  the  beginning  of  any  second,  it  ought  to 
be  drawn  away  from  that  line  j1^  of  an  inch  at  the  end  of 
the  second.  When  the  calculation  was  made  with  the 
correct  distance  of  the  moon,  it  was  found  to  agree  ex- 
actly with  this  result  of  theory.  Thus  it  was  shown  that 
the  force  which  holds  the  moon  in  her  orbit  is  the  same 
which  makes  the  stone  fall,  only  diminished  as  the  inverse 
square  of  the  distance  from  the  centre  of  the  earth.* 

As  it  appeared  that  the  central  forces,  both  toward  the 
sun  and  toward  the  earth,  varied  inversely  as  the  squares 
of  the  distances,  NEWTON  proceeded  to  attack  the  mathe- 
matical problems  involved  in  a  more  systematic  way  than 
any  of  his  predecessors  had  done.  KEPLER'S  second  law 
showed  that  the  line  drawn  from  the  planet  to  the  sun 
wrill  describe  equal  areas  in  equal  times.  NEWTON  showed 
that  this  could  not  be  true,  unless  the  force  which  held 
the  planet  was  directed  toward  the  sun.  We  have  already 
stated  that  the  third  law  showed  that  the  force  was  in- 
versely as  the  square  of  the  distance,  and  thus  agreed  ex- 
actly with  the  theory  of  gravitation.  It  only  remained  to 

*  It  is  a  remarkable  fact  in  the  history  of  science  that  NEWTON 
would  have  reached  this  result  twenty  years  sooner  than  he  did,  had 
he  not  been  misled  by  adopting  an  erroneous  value  of  the  earth's  diame- 
ter. His  first  attempt  to  compute  the  earth's  gravitation  at  the  distance 
of  the  moon  was  made  in  1665,  when  he  was  only  twenty-three  years  of 
age.  At  that  time  he  supposed  that  a  degree  on  the  earth's  surface  was 
sixty  statute  miles,  and  was  in  consequence  led  to  erroneous  results  by 
supposing  the  earth  to  be  smaller  and  the  moon  nearer  than  they  really 
were.  He  therefore  did  not  make  public  his  ideas  ;  but  twenty  years 
later  he  learned  from  the  measures  of  PICARD  in  France  what  the  true 
diameter  of  the  earth  was,  when  he  repeated  his  calculation  with 
entire  success. 


138  ASTRONOMY. 

consider  the  results  of  the  first  law,  that  of  the  elliptic 
motion.  After  long  and  laborious  efforts,  NEWTON  was 
enabled  to  demonstrate  rigorously  that  this  law  also  re- 
sulted from  the  law  of  the  inverse  square,  and  could  result 
from  no  other.  Thus  all  mystery  disappeared  from  the 
celestial  motions  ;  and  planets  were  shown  to  be  simply 
heavy  bodies  moving  according  to  the  same  laws  that  were 
acting  here  around  us,  only  under  very  different  circum- 
stances. All  three  of  KEPLER'S  laws  were  embraced  in 
the  single  law  of  gravitation  toward  the  sun.  The  sun 
attracts  the  planets  as  the  earth  attracts  bodies  here 
around  UB. 

Mutual  Action  of  the  Planets. — It  remained  to  extend 
and  prove  the  theory  by  considering  the  attractions  of  the 
planets  themselves.  By  NEWTON'S  third  law  of  motion, 
each  planet  must  attract  the  sun  with  a  force  equal  to  that 
which  the  sun  exerts  upon  the  planet.  The  moon  also 
must  attract  the  earth  as  much  as  the  earth  attracts  the 
moon.  Such  being  the  case,  it  must  be  highly  probable 
that  the  planets  attract  each  other.  If  so,  KEPLER'S  laws 
can  only  be  an  approximation  to  the  truth.  The  sun, 
being  immensely  more  massive  than  any  of  the  planets, 
overpowers  their  attraction  upon  each  other,  and  makes 
the  law  of  elliptic  motion  very  nearly  true.  But  still  the 
comparatively  small  attraction  of  the  planets  must  cause 
some  deviations.  Now,  deviations  from  the  pure  elliptic 
motion  were  known  to  exist  in  the  case  of  several  of  the 
planets,  notably  in  that  of  the  moon,  which,  if  gravitation 
were  universal,  must  move  under  the  influence  of  the  com- 
bined attraction  of  the  earth  and  of  the  sun.  NEWTON, 
therefore,  attacked  the  complicated  problem  of  the  deter- 
mination of  the  motion  of  the  moon  under  the  combined 
action  of  these  two  forces.  He  showed  in  a  general  way 
that  its  deviations  would  be  of  the  same  nature  as  those 
shown  by  observation.  But  the  complete  solution  of  the 
problem,  which  required  the  answer  to  be  expressed  in 
numbers,  was  beyond  his  power. 


ATTRACTION  OF  GRAVITATION.  139 

Gravitation  Resides  in  each  Particle  of  Matter. — Still 
another  question  arose.  "Were  these  mutually  attractive 
forces  resident  in  the  centres  of  the  several  bodies  attracted, 
or  in  each  particle  of  the  matter  composing  them  ?  NEW- 
TON showed  that  the  latter  must  be  the  case,  because  the 
smallest  bodies,  as  well  as  the  largest,  tended  to  fall 
toward  the  earth,  thus  showing  an  equal  gravitation  in 
every  separate  part.  The  question  then  arose  :  what 
would  be  the  action  of  the  earth  upon  a  body  if  the 
body  was  attracted — not  toward  the  centre  of  the  earth 
alone,  but  toward  every  particle  of  matter  in  the  earth  ? 
It  was  shown  by  a  quite  simple  mathematical  demonstra- 
tion that  if  a  planet  were  on  the  surface  of  the  earth  or 
outside  of  it,  it  would  be  attracted  with  the  same  force  as 
if  the  whole  mass  of  the  earth  were  concentrated  in  its 
centre.  Putting  together  the  various  results  thus  arrived 
at,  NEWTON  was  able  to  formulate  his  great  law  of  uni- 
versal gravitation  in  these  comprehensive  words  :  "  Every 
particle  of  matter  in,  the  universe  attracts  every  other 
particle  with  a  force  directly  as  the  masses  of  the  two 
particles,  and  inversely  as  the  square  of  the  distance 
which  separates  them." 

To  show  the  nature  of  the  attractive  forces  among 
these  various  particles,  let  us  represent  by  m  and  mf  the 
masses  of  two  attracting  bodies.  We  may  conceive  the 
body  m  to  be  composed  of  m  particles,  and  the  other 
body  to  be  composed  of  m'  particles.  Let  us  conceive  that 
each  particle  of  the  one  body  attracts  each  particle  of  the 

other  with  a  force  -a .     Then  every  particle  of  m  will  be 

attracted  by  each  of  the  mf  particles  of  the  other,  and 
therefore  the  total  attractive  force  on  each  of  these  m  par- 
ticles will  be  — .  Each  of  the  m  particles  being  equally 
subject  to  this  attraction,  the  total  attractive  force  between 
the  two  bodies  will  be  — -~.  When  a  given  force  acts 


14:0  ASTRONOMY. 

upon  a  body,  it  will  produce  less  motion  the  larger  the 
body  is,  the  accelerating  force  being  proportional  to  the 
total  attracting  force  divided  by  the  mass  of  the  body 
moved.  Therefore  the  accelerating  force  which  acts  on  the 
body  m' ',  and  which  determines  the  amount  of  motion,  will 

be  — ; ;  and  conversely  the  accelerating  force  acting  on  the 
body  m  will  be  represented  by  the  fraction  -^-. 

§  3.    PROBLEMS  OF  GRAVITATION. 

The  problem  solved  by  NEWTON,  considered  in  its  great- 
est generality,  was  this  :  Two  bodies  of  which  the  masses 
are  given  are  projected  into  space,  in  certain  directions,  and 
with  certain  velocities.  What  will  be  their  motion  under 
the  influence  of  their  mutual  gravitation  ?  If  their  rela- 
tive motion  does  not  exceed  a  certain  definite  amount,  they 
will  each  revolve  around  their  common  centre  of  gravity 
in  an  ellipse,  as  in  the  case  of  planetary  motions.  If,  how- 
ever, the  relative  velocity  exceeds  a  certain  limit,  the  two 
bodies  will  separate  forever,  each  describing  around  the 
common  centre  of  gravity  a  curve  having  infinite  branches. 
These  curves  are  found  to  be  parabolas  in  the  case  where 
the  velocity  is  exactly  at  the  limit,  and  hyperbolas  when 
the  velocity  exceeds  it.  Whatever  curves  may  be  de- 
scribed, the  common  centre  of  gravity  of  the  two  bodies 
will  be  in  the  focus  of  the  curve.  Tims,  when  restricted 
to  two  bodies,  the  problem  admits  of  a  perfectly  rigorous 
mathematical  solution. 

Having  succeeded  in  solving  the  problem  of  planetary 
motion  for  the  case  of  two  bodies,  NEWTON  and  his  son- 
temporaries  very  naturally  desired  to  effect  a  similar  solu- 
tion for  the  case  of  three  bodies.  The  problem  of  motion 
in  our  solar  system  is  that  of  the  mutual  action  of  a  great 
number  of  bodies  ;  and  Laving  succeeded  in  the  case  of 
two  bodies,  it  was  necessary  next  to  try  that  of  three. 


PROBLEMS  OF  GRAVITATION.  141 

Thus  arose  the  celebrated  problem  of  three  bodies.  It  is 
found  that  no  rigorous  and  general  solution  of  this  problem 
is  possible.  The  curves  described  by  the  several  bodies 
would,  in  general,  be  so  complex  as  to  defy  mathematical 
definition.  But  in  the  special  case  of  motions  in  the  solar 
system,  the  problem  admits  of  being  solved  by  approxima- 
tion with  any  required  degree  of  accuracy.  The  princi- 
ples involved  in  this  system  of  approximation  may  be  com- 
pared to  those  involved  in  extracting  the  square  root  of 
any  number  which  is  not  an  exact  square  ;  2  for  instance. 
The  square  root  of  2  cannot  be  exactly  expressed  either 
by  a  decimal  or  vulgar  fraction  ;  but  by  increasing  the 
number  of  figures  it  can  be  expressed  to  any  required  limit 
of  approximation.  Thus,  the  vulgar  fractions  f ,  \ },  |^-J, 
etc. ,  are  fractions  which  approach  more  and  more  to  the 
required  quantity  ;  and  by  using  larger  numbers  the  errors 
of  such  fraction  may  be  made  as  small  as  we  please.  So,  in 
using  decimals,  we  diminish  the  error  by  one  tenth  for  eve- 
ry decimal  we  add,  but  never  reduce  it  to  zero.  A  process 
of  the  same  nature,  but  immensely  more  complicated,  has 
to  be  used  in  computing  the  motions  of  the  planets  from 
their  mutual  gravitation.  The  possibility  of  such  an  ap- 
proximation arises  from  the  fact  that  the  planetary  orbits 
are  nearly  circular,  and  that  their  masses  are  very  small 
compared  with  that  of  the  sun.  The  first  approximation 
is  that  of  motion  in  an  ellipse.  In  this  way  the  motion  of 
a  planet  through  several  revolutions  can  nearly  always  be 
predicted  within  a  small  fraction  of  a  degree,  though  it 
may  wander  widely  in  the  course  of  centuries.  Then  sup- 
pose each  planet  to  move  in  a  known  ellipse  ;  their  mutual 
attraction  at  each  point  of  their  respective  orbits  can  be 
expressed  by  algebraic  formulae.  In  constructing  these 
formulae,  the  orbits  are  first  supposed  to  be  circular  ;  and 
afterward  account  is  taken  by  several  successive  steps  of 
the  eccentricity.  Having  thus  found  approximately  their 
action  on  each  other,  the  deviations  from  the  pure  elliptic 
motion  produced  by  this  action  may  be  approximately  cal- 


ASTRONOMY. 

culated.  This  being  done,  the  motions  will  be  more  exact- 
ly determined,  and  the  mutual  action  can  be  more  exactly 
calculated.  Thus,  the  process  can  be  carried  on  step  by 
step  to  any  degree  of  precision  ;  but  an  enormous  amount 
of  calculation  is  necessary  to  satisfy  the  requirements  of 
modern  times  with  respect  to  precision.*  As  a  general 
rule,  every  successive  step  in  the  approximation  is  much 
more  laborious  than  all  the  preceding  ones. 

To  understand  the  principle  of  astronomical  investiga- 
tion into  the  motion  of  the  planets,  the  distinction  be- 
tween observed  and  theoretical  motions  must  be  borne  in 
mind.  When  the  astronomer  with  his  meridian  circle  de- 
termines the  position  of  a  planet  on  the  celestial  sphere, 
that  position  is  an  observed  one.  When  he  calculates  it,  for 
the  same  instant,  from  theory,  or  from  tables  founded  on 
the  theory,  the  result  will  be  a  calculated  or  theoretical 
position.  The  two  are  to  be  regarded  as  separate,  no  mat- 
ter if  they  should  be  exactly  the  same  in  reality,  because 
they  have  an  entirely  different  origin.  But  it  must  be  re- 
membered that  no  position  can  be  calculated  from  theory 
alone  independent  of  observation,  because  all  sound  theory 
requires  some  data  to  start  with,  which  observation  alone 
can  furnish.  In  the  case  of  planetary  motions,  these  data 
are  the  elements  of  the  planetary  orbit  already  described, 
or,  which  amounts  to  the  same  tiling,  the  velocity  and  di- 
rection of  the  motion  of  the  planet  as  well  as  its  mass  at 
some  given  time.  If  these  quantities  were  once  given 
with  mathematical  precision,  it  would  be  possible,  from  the 
theory  of  gravitation  alone,  without  recourse  to  observa- 
tion, to  predict  the  motions  of  the  planets  day  by  day 
and  generation  after  generation  with  any  required  degree 
of  precision,  always  supposing  that  they  are  subjected  to  no 
influence  except  their  mutual  gravitation  according  to  the 
law  of  NEWTON.  But  it  is  impossible  to  determine  the 
elements  or  the  velocities  without  recourse  to  observation  ; 

*  In  the  works  of  the  great  mathematicians  on  this  subject,  algebraic 
formulae  extending  through  many  pages  are  sometimes  gi\ren. 


PROBLEMS  OF  GRAVITATION.  143 

and  however  correctly  they  may  seemingly  be  determined 
for  the  time  being,  subsequent  observations  always  show 
them  to  have  been  more  or  less  in  error.  The  reader 
must  understand  that  no  astronomical  observation  can  be 
mathematically  exact.  Both  the  instruments  and  the 
observer  are  subjected  to  influences  which  prevent  more 
than  an  approximation  being  attained  from  any  one 
observation.  The  great  art  of  the  astronomer  consists  in 
so  treating  and  combining  his  observations  as  to  eliminate 
their  errors,  and  give  a  result  as  near  the  truth  as  possible. 
When,  by  thus  combining  his  observations,  the  astrono- 
mer has  obtained  the  elements  of  the  planet's  motion  which 
he  considers  to  be  near  the  truth,  he  calculates  from  them 
a  series  of  positions  of  the  planet  from  day  to  day  in  the 
future,  to  be  compared  with  subsequent  observations.  If 
he  desires  his  work  to  be  more  permanent  in  its  nature, 
he  may  construct  tables  by  which  the  position  can  be  de- 
termined at  any  future  time.  Having  thus  a  series  of  the- 
oretical or  calculated  places  of  the  planet,  he,  or  others, 
will  compare  his  predictions  with  observation,  and  from 
the  differences  deduce  corrections  to  his  elements.  We 
may  say  in  a  rough  way  that  if  a  planet  has  been  observed 
through  a  certain  number  of  years,  it  is  possible  to  calculate 
its  place  for  an  equal  number  of  years  in  advance  with 
some  approach  to  precision.  Accurate  observations  are 
commonly  supposed  to  commence  with  BRADLEY,  Astron- 
omer Hoyal  of  England  in  1750.  A  century  and  a  quarter 
having  elapsed  since  that  time,  it  is  now  possible  to  con- 
struct tables  of  the  planets,  which  we  may  expect  to  be 
tolerably  accurate,  until  the  year  2000.  But  this  is  a 
possibility  rather  than  a  reality.  The  amount  of  calcu- 
lation required  for  such  wTork  is  so  immense  as  to  be  en- 
tirely beyond  the  power  of  any  one  person,  and  hence  it  is 
only  when  a  mathematician  is  able  to  command  the  ser- 
vices of  others,  or  when  several  mathematicians  in  some 
way  combine  for  an  object,  that  the  best  astronomical 
tables  can  hereafter  be  constructed. 


144  ASTRONOMY. 

§  4.    RESULTS  OP  GRAVITATION. 

From  what  we  have  said,  it  will  be  seen  that  the  problem 
of  the  motions  of  the  planets  under  the  influence  of  grav- 
itation has  called  forth  all  the  skill  of  the  mathematicians 
who  have  attacked  it.  They  actually  find  themselves  able 
to  reach  a  solution,  which,  so  far  as  the  mathematics  of  the 
subject  are  concerned,  may  be  true  for  many  centuries,  but 
not  a  solution  which  shall  be  true  for  all  time.  Among 
those  who  have  brought  the  solution  so  near  to  perfec- 
tion, LA  PLACE  is  entitled  to  the  first  rank,  although  there 
are  others,  especially  LA  GRANGE,  who  are  fully  worthy  to 
be  named  along  with  him.  It  will  be  of  interest  to  state 
the  general  results  reached  by  these  and  other  mathema- 
ticians. 

We  call  to  mind  that  but  for  the  attraction  of  the 
planets  upon  each  other,  every  planet  would  move  around 
the  sun  in  an  invariable  ellipse,  according  to  KEPLER'S 
laws.  The  deviations  from  this  elliptic  motion  produced 
by  their  mutual  attraction  are  called  perturbations.  When 
they  were  investigated,  it  was  found  that  they  were  of  two 
classes,  which  were  denominated  respectively  periodic 
perturbations  and  secular  variations. 

The  periodic  perturbations  consist  of  oscillations  depend- 
ent upon  the  mutual  positions  of  the  planets,  and  there- 
fore of  comparatively  short  period.  Whenever,  after  a 
number  of  revolutions,  two  planets  return  to  the  same 
position  in  their  orbits,  the  periodic  perturbations  are  of 
the  same  amount  so  far  as  these  two  planets  are  concerned. 
They  may  therefore  be  algebraically  expressed  as  depend- 
ent upon  the  longitude  of  the  two  planets,  the  disturbing 
one  and  the  disturbed  one.  For  instance,  the  perturba- 
tions of  the  earth  produced  by  the  action  of  Mercury 
depend  on  the  longitude  of  the  earth  and  on  that  of  Mer- 
cury. Those  produced  by  the  attraction  of  Venus  de- 
pend upon  the  longitude  of  the  earth  and  on  that  of 
Venus,  and  so  on. 


RESULTS  OF  GRAVITATION.  145 

The  secular  perturbations )  or  secular  variations  as  they 
are  commonly  called,  consist  of  slow  changes  in  the  forms 
and  positions  of  the  several  orbits.  It  is  found  that  the 
perihelia  of  all  the  orbits  are  slowly  changing  their  ap- 
parent directions  from  the  sun  ;  that  the  eccentricities  of 
some  are  increasing  and  of  others  diminishing  ;  and  that 
the  positions  of  the  orbits  are  also  changing. 

One  of  the  first  questions  which  arose  in  reference  to 
these  secular  variations  was,  will  they  go  on  indefinitely  ? 
If  they  should,  they  would  evidently  end  in  the  subversion 
of  the  solar  system  and  the  destruction  of  all  life  upon  the 
earth.  The  orbits  of  the  earth  and  planets  would,  in  the 
course  of  ages,  become  so  eccentric,  that,  approaching 
near  the  sun  at  one  time  and  receding  far  away  from  it  at 
another,  the  variations  of  temperature  would  be  destruc- 
tive to  life.  This  problem  was  first  solved  by  LA  GRANGE. 
He  showed  that  the  changes  could  not  go  on  forever,  but 
that  each  eccentricity  would  always  be  confined  between 
two  quite  narrow  limits.  His  results  may  be  expressed 
by  a  very  simple  geometrical  construction.  Let  S  repre- 
sent the  sun  situated  in  the  focus  of  the  ellipse  in  which 


PIG.  54. 

the  planet  moves,  and  let  0  be  the  centre  of  the  ellipse. 
Let  a  straight  line  SB  emanate  from  the  sun  to  B, 
another  line  pass  from  B  to  D,  and  so  on  ;  the  number  of 
these  lines  being  equal  to  that  of  the  planets,  and  the  last 
one  terminating  in  C,  the  centre  of  the  ellipse.  Then  the 
line  S  B  will  be  moving  around  the  sun  with  a  very  slow 
motion  ;  B  D  will  move  around  B  with  a  slow  motion 
somewhat  different,  and  so  each  one  will  revolve  in  the 


146  ASTRONOMY. 

same  manner  until  we  reach  the  line  which  carries  on  its 
end  the  centre  of  the  ellipse.  These  motions  are  so  slow 
that  some  of  them  require  tens  of  thousands,  and  others 
hundreds  of  thousands  of  years  to  perform  the  revolution. 
By  the  combined  motion  of  them  all,  the  centre  of  the 
ellipse  describes  a  somewhat  irregular  curve.  It  is  evi 
dent,  however,  that  the  distance  of  the  centre  from  the 
sun  can  never  be  greater  than  the  sum  of  these  revolving 
lines.  Now  this  distance  shows  the  eccentricity  of  the 
ellipse,  which  is  equal  to  half  the  difference  between  the 
greatest  and  least  distances  of  the  planet  from  the  sun. 
The  perihelion  being  in  the  direction  C  S,  on  the  opposite 
side  of  the  sun  from  (7,  it  is  evident  that  the  motion  of 
C  will  carry  the  perihelion  with  it.  It  is  found  in  this 
way  that  the  eccentricity  of  the  earth's  orbit  has  been 
diminishing  for  about  eighteen  thousand  years,  and  will 
continue  to  diminish  for  twenty-five  thousand  years  to 
come,  when  it  will  be  more  nearly  circular  than  any  orbit 
of  our  system  now  is.  But  before  becoming  quite  circu- 
lar, the  eccentricity  will  begin  to  increase  again,  and  so  go 
on  oscillating  indefinitely. 

Secular  Acceleration  of  the  Moon. —  Another  remark- 
able result  reached  by  mathematical  research  is  that  of  the 
acceleration  of  the  moon's  motion.  More  than  a  century 
ago  it  was  found,  by  comparing  the  ancient  and  modern 
observations  of  the  moon,  that  the  latter  moved  around  the 
earth  at  a  slightly  greater  rate  than  she  did  in  ancient 
times.  The  existence  of  this  acceleration  was  a  source  of 
great  perplexity  to  LA  GRANGE  and  LA  PLACE,  because 
they  thought  that  they  had  demonstrated  mathematically 
that  the  attraction  could  not  have  accelerated  or  retarded 
the  mean  motion  of  the  moon.  But  on  continuing  his  in- 
vestigation, LA  PLACE  found  that  there  was  one  cause 
which  he  omitted  to  take  account  of — namely,  the  secular 
diminution  in  the  eccentricity  of  the  earth's  orbit,  of 
which  we  have  just  spoken.  He  found  that  this  change 
in  the  eccentricity  would  slightly  alter  the  action  of  the 


ACCELERATION  01    THE  MOON.  14? 

sun  upon  the  moon,  and  that  this  alteration  of  action 
would  be  such  that  so  long  as  the  eccentricity  grew 
smaller,  the  motion  of  the  moon  would  continue  to  be  ac- 
celerated. Computing  the  moon's  acceleration,  he  found  it 
to  be  equal  to  ten  seconds  into  the  square  of  the  number 
of  centuries,  the  law  being  the  same  as  that  for  the  motion 
of  a  falling  body.  That  is,  while  in  one  century  she  would 
be  ten  seconds  ahead  of  the  place  she  would  have  occupied 
had  her  mean  motion  been  uniform,  she  would,  in  two 
centuries,  be  forty  seconds  ahead,  in  three  centuries  ninety 
seconds,  and  so  on  ;  and  during  the  two  thousand  years 
which  have  elapsed  since  the  observations  of  HIPP  ARCH  us, 
the  acceleration  would  be  more  than  a  degree.  It  has  re- 
cently been  found  that  LA  PLACE'S  calculation  was  not  com- 
plete, and  that  with  the  more  exact  methods  of  recent  times 
the  real  acceleration  computed  from  the  theory  of  gravita- 
tion is  only  about  six  seconds.  The  observations  of  ancient 
eclipses,  however,  compared  with  our  modern  tables,  show 
an  acceleration  greater  than  this  ;  but  owing  to  the  rude 
and  doubtful  character  of  nearly  all  the  ancient  data,  there 
is  some  doubt  about  the  exact  amount.  From  the  most 
celebrated  total  eclipses  of  the  sun,  an  acceleration  of  about 
twelve  seconds  is  deduced,  while  the  observations  of 
PTOLEMY  and  the  Arabian  astronomers  indicate  only  eight 
or  nine  seconds.  There  is  thus  an  apparent  discrepancy 
between  theory  and  observation,  the  latter  giving  a  larger 
value  to  the  acceleration.  This  difference  is  now  accounted 
for  by  supposing  that  the  motion  of  the  earth  on  its  axis 
is  retarded — that  is,  that  the  day  is  gradually  growing 
longer.  From  the  modern  theory  of  friction,  it  is  found 
that  the  motion  of  the  ocean  under  the  influence  of  the 
moon's  attraction  which  causes  the  tides,  must  be  accom- 
panied with  some  friction,  and  that  this  friction  must  re- 
tard the  earth's  rotation.  There  is,  however,  no  way  of 
determining  the  amount  of  this  retardation  unless  we 
assume  that  it  causes  the  observed  discrepancy  between 
the  theoretical  and  observed  accelerations  of  the  moon. 


148  ASTHOXOMY. 

How  this  effect  is  produced  will  be  seen  by  reflecting  that 
if  the  day  is  continually  growing  longer  without  our  know- 
ing it,  our  observations  of  the  moon,  which  we  may  suppose 
to  be  made  at  noon,  for  example,  will  be  constantly  made  a 
little  later,  because  the  interval  from  one  noon  to  another 
will  be  continually  growing  a  little  longer.  The  moon  con- 
tinually moving  forward,  the  observation  will  place  her  fur- 
ther and  further  ahead  than  she  would  have  been  observed 
had  there  been  no  retardation  of  the  time  of  noon.  If  in 
the  course  of  ages  our  noon-dials  get  to  be  an  hour  too 
late,  we  should  find  the  moon  ahead  of  her  calculated  place 
by  one  hour's  motion,  or  about  a  degree.  The  present 
theory  of  acceleration  is,  therefore,  that  the  moon  is  really 
accelerated  about  six  seconds  in  a  century,  and  that  the 
motion  of  the  earth  on  its  axis  is  gradually  diminishing 
at  such  a  rate  as  to  produce  an  apparent  additional  ac- 
celeration which  may  range  from  two  to  six  seconds. 


§  5.    REMARKS    ON  THE    THEORY    OF    GRAVITA- 
TION. 

The  real  nature  of  the  great  discovery  of  NEWTON  is  so 
frequently  misunderstood  that  a  little  attention  may  be 
given  to  its  elucidation.  Gravitation  is  frequently  spoken 
of  as  if  it  were  a  theory  of  NEWTON'S,  and  very  generally 
received  by  astronomers,  but  still  liable  to  be  ultimately 
rejected  as  a  great  many  other  theories  have  been.  Not 
infrequently  people  of  greater  or  less  intelligence  are 
found  making  great  efforts  to  prove  it  erroneous.  Every 
prominent  scientific  institution  in  the  world  frequently 
receives  essays  having  this  object  in  view.  Now,  the  fact 
is  that  NEWTON  did  not  discover  any  new  force,  but  only 
showed  that  the  motions  of  the  heavens  could  be  accounted 
for  by  a  force  which  we  all  know  to  exist.  Gravitation 
(Latin  gravitas — weight,  heaviness)  is,  properly  speaking, 
the  force  which  makes  all  bodies  here  at  the  surface  of  the 
earth  tend  to  fall  downward  ;  and  if  any  one  wishes  to 


REALITY  OF  GRAVITATION.  140 

subvert  the  theory  of  gravitation,  he  must  begin  by  prov- 
ing that  this  force  does  not  exist.  This  no  one  would 
think  of  doing.  What  NEWTON  did  was  to  show  that 
this  force,  which,  before  his  time,  had  been  recognized 
only  as  acting  on  the  surface  of  the  earth,  really  extended 
to  the  heavens,  and  that  it  resided  not  only  in  the  earth 
itself,  but  in  the  heavenly  bodies  also,  and  in  each  particle 
of  matter,  however  situated.  To  put  the  matter  in  a  terse 
form,  what  NEWTON  discovered  was  not  gravitation,  but 
the  universality  of  gravitation. 

It  may  be  inquired,  is  the  induction  which  supposes 
gravitation  universal  so  complete  as  to  be  entirely  beyond 
doubt  ?  We  reply  that  within  the  solar  system  it  certainly 
is.  The  laws  of  motion  as  established  by  observation  and 
experiment  at  the  surface  of  the  earth  must  be  considered 
as  mathematically  certain.  Now,  it  is  an  observed  fact 
that  the  planets  in  their  motions  deviate  from  straight 
lines  in  a  certain  way.  By  the  first  law  of  motion,  such 
deviation  can  be  produced  only  by  a  force  ;  and  the  direc- 
tion and  intensity  of  this  force  admit  of  being  calculated 
once  that  the  motion  is  determined.  When  thus  calculated, 
it  is  found  to  be  exactly  represented  by  one  great  force 
constantly  directed  toward  the  sun,  and  smaller  subsidiary 
forces  directed  toward  the  several  planets.  Therefore, 
no  fact  in  nature  is  more  firmly  established  than  is  that  of 
universal  gravitation,  as  laid  down  by  NEWTON,  at  least 
within  the  solar  system. 

We  shall  find,  in  describing  double  stars,  that  gravita- 
tion is  also  found  to  act  between  the  components  of  a  great 
number  of  such  stars.  It  is  certain,  therefore,  that  at 
least  some  stars  gravitate  toward  each  other,  as  the  bodies 
of  the  solar  system  do  ;  but  the  distance  which  separates 
most  of  the  stars  from  each  other  and  from  our  sun  is  so 
immense  that  no  evidence  of  gravitation  between  them 
has  yet  been  given  by  observation.  Still,  that  they  do 
gravitate  according  to  NEWTON'S  law  can  hardly  be  seri- 
ously doubted  by  any  one  who  understands  the  subject. 


150  ASTRONOMY. 

The  reader  may  now  be  supposed  to  see  the  absurdity  of 
supposing  that  the  theory  of  gravitation  can  ever  be  sub- 
verted. It  is  not,  however,  absurd  to  suppose  that  it  may 
yet  be  shown  to  be  the  result  of  some  more  general  law. 
Attempts  to  do  this  are  made  from  time  to  time  by  men 
of  a  philosophic  spirit  ;  but  thus  far  no  theory  of  the  sub- 
ject having  the  slightest  probability  in  its  favor  has  been 
propounded. 

Perhaps  one  of  the  most  celebrated  of  these  theories  is 
that  of  GEORGE  LEWIS  LE  SAGE,  a  Swiss  physicist  of  the 
last  century.  He  supposed  an  infinite  number  of  ultra- 
mundane corpuscles,  of  transcendent  minuteness  and  veloc- 
ity, traversing  space  in  straight  lines  in  all  directions.  A 
single  body  placed  in  the  midst  of  such  an  ocean  of  mov- 
ing corpuscles  would  remain  at  rest,  since  it  would  be  equal- 
ly impelled  in  every  direction.  But  two  bodies  would  ad- 
vance toward  each  other,  because  each  of  them  would 
screen  the  other  from  these  corpuscles  moving  in  the 
straight  line  joining  their  centres,  and  there  would  be  a 
slight  excess  of  corpuscles  acting  on  that  side  of  each 
body  which  was  turned  away  from  the  other.* 

One  of  the  commonest  conceptions  to  account  for  grav- 
itation is  that  of  a  fluid,  or  ether,  extending  through  all 
space,  which  is  supposed  to  be  animated  by  certain  vibra- 
tions, and  forms  a  vehicle,  as  it  were,  for  the  transmission 
of  gravitation.  This  and  all  other  theories  of  the  kind 
are  subject  to  the  fatal  objection  of  proposing  complicated 
systems  to  account  for  the  most  simple  and  elementary 
facts.  If,  indeed,  such  systems  were  otherwise  known  to 
exist,  and  if  it  could  be  shown  that  they  really  would 
produce  the  effect  of  gravitation,  they  would  be  entitled 
to  reception.  But  since  they  have  been  imagined  only  to 
account  for  gravitation  itself,  and  since  there  is  no  proof 
of  their  existence  except  that  of  accounting  for  it,  they 

*  Reference  may  be  made  to  an  article  on  the  kinetic  theories  of 
gravitation  by  William  B.  Taylor,  in  the  Smithsonian  Report  for 

1876. 


CAUSE  OF  GRAVITATION.  151 

are  not  entitled  to  any  weight  whatever.  In  the  present 
state  of  science,  we  are  justified  in  regarding  gravitation  as 
an  ultimate  principle  of  matter,  incapable  of  alteration  by 
any  transformation  to  which  matter  can  be  subjected. 
The  most  careful  experiments  show  that  no  chemical  pro- 
cess to  which  matter  can  be  subjected  either  increases  or 
diminishes  its  gravitating  principles  in  the  slightest  degree. 
We  cannot  therefore  see  how  this  principle  can  ever  be 
referred  to  any  more  general  cause. 


CHAPTER  VI. 

THE  MOTIONS  AND  ATTRACTION  OP  THE  MOON. 

EACH  of  the  planets,  except  Mercury  and  Venus,  is  at  - 
tended  by  one  or  more  satellites,  or  moons  as  they  are  some- 
times familiarly  called.  These  objects  revolve  around  their 
several  planets  in  nearly  circular  orbits,  accompanying  them 
in  their  revolutions  around  the  sun.  Their  distances  from 
their  planets  are  very  small  compared  with  the  distances 
of  the  latter  from  each  other  and  from  the  sun.  Their 
magnitudes  also  are  very  small  compared  with  those  of  the 
planets  around  which  they  revolve.  Where  there  are 
several  satellites  revolving  around  a  planet,  the  whole  of 
these  bodies  forms  a  small  system  similar  to  the  solar  sys- 
tem in  arrangement.  Considering  each  system  by  itself, 
the  satellites  revolve  around  their  central  planets  or 
i  i  primaries, ' '  in  nearly  circular  orbits,  much  as  the  planets 
revolve  around  the  sun.  But  each  system  is  carried  around 
the  sun  without  any  serious  derangement  of  the  motion 
of  its  several  bodies  among1  themselves. 

Our  earth  has  a  single  satellite  accompanying  it  in  this 
way,  the  familiar  moon.  It  revolves  around  the  earth  in 
a  little  less  than  a  month.  The  nature,  causes  and  con- 
sequences of  this  motion  form  the  subject  of  the  present 
chapter. 

§   1.     THE    MOON'S    MOTIONS    AND    PHASES. 

That  the  moon  performs  a  monthly  circuit  in  the  heav- 
ens is  a  fact  with  which  wTe  are  all  familiar  from  child- 
hood. At  certain  times  we  see  her  newly  emerged  from 


MOTION  OF  THE  MOON.  153 

the  snn's  rays  in  the  western  twilight,  and  then  we  call 
her  the  new  moon.  On  each  succeeding  evening,  we  see 
her  further  to  the  east,  so  that  in  two  weeks  she  is  oppo- 
site the  sun,  rising  in  the  east  as  he  sets  in  the  west. 
Continuing  her  course  two  weeks  more,  she  has  approached 
the  sun  on  the  other  side,  or  from  the  west,  and  is  once 
more  lost  in  his  rays.  At  the  end  of  twenty-nine  or  thirty 
days,  we  see  her  again  emerging  as  new  moon,  and  her  cir- 
cuit is  complete.  It  is,  however,  to  be  remembered 
that  the  sun  has  been  apparently  moving  toward  the  east 
among  the  stars  during  the  whole  month,  so  that  during 
the  interval  from  one  new  moon  to  the  next  the  moon  has 
to  make  a  complete  circuit  relatively  to  the  stars,  and 
move  forward  some  30°  further  to  overtake  the  sun.  The 
revolution  of  the  moon  among  the  stars  is  performed  in 
about  27-J  days,*  so  that  if  we  observe  when  the  moon  is 
very  near  some  star,  we  shall  find  her  in  the  same  position 
relative  to  the  star  at  the  end  of  this  interval. 

The  motion  of  the  moon  in  this  circuit  differs  from  the 
apparent  motions  of  the  planets  in  being  always  forward. 
"We  have  seen  that  the  planets,  though,  on  the  whole,  mov- 
ing directly,  or  toward  the  east,  are  affected  with  an  ap- 
parent retrograde  motion  at  certain  intervals,  owing  to  the 
motion  of  the  earth  around  the  sun.  But  the  earth  is  the 
real  centre  of  the  moon's  motion,  and  carries  the  moon 
along  with  it  in  its  annual  revolution  around  the  sun.  To 
form  a  correct  idea  of  the  real  motion  of  these  three 
bodies,  we  must  imagine  the  earth  performing  its  circuit 
around  the  sun  in  one  year,  and  carrying  with  it  the  moon, 
which  makes  a  revolution  around  it  in  27  days,  at  a  distance 
only  about  ^-J-^  that  of  the  sun. 

In  Fig.  55  suppose  S  to  represent  the  sun,  the  large 
circle  to  represent  the  orbit  of  the  earth  around  it,  E  to 
be  some  position  of  the  earth,  and  the  dotted  circle  to  rep- 
resent the  orbit  of  the  moon  around  the  earth.  We  must 

*  More  exactly,  27d  32166. 


154 


ASTRONOMY. 


is  at  E  the  moon  is  at  M. 


FIG.  55. 


imagine  the  latter  to  carry  this  circle  with  it  in  its  an- 
nual course  around  the  sun.     Suppose  that  when  the  earth 

Then  if  the  earth  move  to 
EI  in  27£  days,  the  moon 
will  have  made  a  complete 
revolution  relative  to  the 
stars — that  is,  it  will  be  at 
MV  the  line  El  M^  being  par- 
allel to  EM.  But  new 
moon  will  not  have  arrived 
again  because  the  sun  is  not 
in  the  same  direction  as  be- 
fore. The  moon  must  move 
through  the  additional  arc 
Ml  EM»  and  a  little  more, 
owing  to  the  continual  ad- 
vance of  the  earth,  before  it 
will  again  be  new  moon. 
Phases  of  the  Moon. — The  moon  being  a  non -luminous 
body  shines  only  by  reflecting  the  light  falling  on  her 
from  some  other  body.  The  principal  source  of  light  is 
the  sun.  Since  the  moon  is  spherical  in  shape,  the  sun 
can  illuminate  one  half  her  surface.  The  appearance  of 
the  moon  varies  according  to  the  amount  of  her  illumi- 
nated hemisphere  which  is  turned  toward  the  earth,  as 
can  be  seen  by  studying  Fig.  56.  Here  the  central 
globe  is  the  earth  ;  the  circle  around  it  represents  the  orbit 
of  the  moon.  The  rays  of  the  sun  fall  on  both  earth  and 
moon  from  the  right,  the  distance  of  the  sun  being,  on  the 
scale  of  the  figure,  some  30  feet.  Eight  positions  of  the 
moon  are  shown  around  the  orbit  at  A,  E,  C,  etc.,  and 
the  right-hand  hemisphere  of  the  moon  is  illuminated  in 
each  position.  Outside  these  eight  positions  are  eight 
others  showing  how  the  moon  looks  as  seen  from  the  earth 
in  each  position. 

At   A   it   is    "new   moon,"    the   moon   being  nearly 
between    the  earth   and   the  sun.     Its  dark   hemisphere 


PHASES  OF  THE  MOON. 


155 


is  then  turned  toward  the  earth,   so   that   it  is  entirely 
invisible. 

At  ^the  observer  on  the  earth  sees  about  a  fourth  of 
the  illuminated  hemisphere,  which  looks  like  a  crescent, 
as  shown  in  the  outside  figure.  In  this  position  a  great 
deal  of  light  is  reflected  from  the  earth  to  the  moon,  ren- 
dering the  dark  part  of  the  latter  visible  by  a  gray  light. 


FIG.  56. 

This  appearance  is  sometimes  called  the  "  old  moon  in 
the  new  moon's  arms." 

At  C  the  moon  is  said  to  be  in  her  "  first  quarter,"  and 
one  half  her  illuminated  hemisphere  is  visible. 

At  G  three  fourths  of  the  illuminated  hemisphere  is 
visible,  and  at  B  the  whole  of  it.  The  latter  position,  when 
the  moon  is  opposite  the  sun,  is  called  "  full  moon." 

After  this,  at  H,  Z>,  f\  the  same  appearances  are  re- 
peated in  the  reversed  order,  the  position  D  being  called 
the  c '  last  quarter. ' ' 


156  ASTRONOMY. 

The  four  principal  phases  of  the  moon  are,  "  New 
moon,"  "  First  quarter,"  "  Full  moon,"  "  Last  quarter," 
which  occur  in  regular  and  unending  succession,  at  inter- 
vals of  between  7  and  8  days. 

§2.    THE    SUN'S    DISTURBING    FORCE. 

The  distances  of  the  sun  and  planets  being  so  immensely 
great  compared  with  that  of  the  moon,  their  attraction 
upon  the  earth  and  the  moon  is  at  all  times  very  nearly 
equal.  Now  it  is  an  elementary  principle  of  mechanics 
that  if  two  bodies  are  acted  upon  by  equal  and  parallel 
forces,  no  matter  how  great  these  forces  may  be,  the 
bodies  will  move  relatively  to  each  other  as  if  those  forces 
did  not  act  at  all,  though  of  course  the  absolute  motion  of 
each  will  be  different  from  what  it  otherwise  would  be. 
If  we  calculate  the  absolute  attraction  of  the  sun  upon  the 
moon,  we  shall  find  it  to  be  about  twice  as  great  as  that  of 
the  earth,  because,  although  it  is  situated  at  400  times  the 
distance,  its  mass  is  about  330,000  times  as  great  as  that  of 
the  earth,  and  if  we  divide  this  mass  by  the  square  of  the 
distance  400  we  have  2  as  the  quotient. 

To  those  unacquainted  with  mechanics,  the  difficulty 
often  suggests  itself  that  the  sun  ought  to  draw  the  moon 
away  from  the  earth  entirely.  But  we  are  to  remember 
that  the  sun  attracts  the  earth  in  the  same  way  that  it  at- 
tracts the  moon,  so  that  the  difference  between  the  sun's 
attraction  on  the  moon  and  on  the  earth  is  only  a  small 
fraction  of  the  attraction  between  the  earth  and  the  moon .  * 

As  a  consequence  of  these  forces,  the  moon  moves  around 
the  earth  nearly  as  if  neither  of  them  were  attracted  by 

*  In  this  comparison  of  the  attractive  forces  of  the  sun  upon  the 
moon  and  upon  the  earth,  the  reader  will  remember  that  we  are  speak- 
ing not  of  the  absolute  force,  but  of  what  is  called  the  accelerating  force, 
which  is  properly  the  ratio  of  the  absolute  force  to  the  mass  of  the 
body  attracted.  The  earth  having  80  times  the  mass  of  the  moon,  the 
sun  must  of  course  attract  it  with  80  times  the  absolute  force  in  order 
to  produce  the  same  motion,  or  the  same  accelerating  force. 


SUN'S  ATTRACTION  ON  MOON.  157 

the  sun — that  is,  nearly  in  an  ellipse,  having  the  earth  in 
its  focus.  But  there  is  always  a  small  difference  between 
the  attractive  forces  of  the  sun  upon  the  moon  and  upon  the 
earth,  and  this  difference  constitutes  a  disturbing  force 
which  makes  the  moon  deviate  from  the  elliptic  orbit 
which  it  would  otherwise  describe,  and,  in  fact,  keeps  the 
ellipse  which  it  approximately  describes  in  a  state  of  con- 
stant change. 

A  more  precise  idea  of  the  manner  in  which  the  sun  disturbs  the 
motion  of  the  moon  around  the  earth  may  be  gathered  from 
Fig.  57.  Here  S  represents  the  sun,  and  the  circle  F  Q  M  N repre- 
sents the  orbit  of  the  moon.  First  suppose  the  moon  at  JV,  the  posi- 
tion corresponding  to  new  moon.  Then  the  moon,  being  nearer  to 
the  sun  than  the  earth  is,  will  be  attracted  more  powerfully  by  it 
than  the  earth  is.  It  will  therefore  be  drawn  away  from  the  earth, 
or  the  action  of  the  sun  will  tend  to  separate  the  two  bodies. 


FIG.  57. 

Next  suppose  the  moon  at  F  the  position  corresponding  to  full 
moon.  Here  the  action  of  the  sun  upon  the  earth  will  be  more 
powerful  than  upon  the  moon,  and  the  earth  will  in  consequence  be 
drawn  away  from  the  moon.  In  this  position  also  the  effect  of  the 
disturbing  force  is  to  separate  the  two  bodies.  If,  on  the  other 
hand,  the  moon  is  near  the  first  quarter  or  near  Q,  the  sun  will  exert 
a  nearly  equal  attraction  on  both  bodies  ;  and  ince  the  lines  of  at- 
traction E  S  and  Q  8  then  converge  toward  St  it  follows  that  there 
will  be  a  tendency  to  bring  the  two  bodies  together.  The  same 
will  evidently  be  true  at  the  third  quarter.  Hence  the  influence  of 
the  disturbing  force  changes  back  and  forth  twice  in  the  course  of 
each  lunar  month. 

The  disturbing  force  in  question  may  be  constructed  for  any  po- 
sition of  the  moon  in  its  orbit  in  the  following  way,  which  is  be- 
lieved to  be  due  to  Mr.  R.  A.  PROCTOR  :  Let  M  be  the  position  of 
the  moon  ;  let  us  represent  the  sun's  attraction  upon  it  by  the  line 
M  S,  and  let  us  investigate  what  line  will  represent  the  sun's  attrac- 
tion upon  the  earth  on  the  same  scale.  From  M  drop  the  perpen- 


158  ASTRONOMY. 

dicular  M  P  upon  the  line  E  8  joining  the  sun  to  the  earth.  This 
attraction  being  inversely  as  the  square  of  the  distance,  we  shall 
have, 

Attraction  on  earth  _  SM* 

Attraction  on  moon  ~~  S  E*' 

We  have  taken  the  line  S  M  itself  to  represent  the  attraction  on 
the  moon,  so  that  we  have 

Attraction  on  moon  =  SM. 
Multiplying  the  two  equations  member  by  member,  we  find, 

O  \f'i 


Attraction  on  earth  =  8  M  x 


-^  —  -. 


The  line  S  M  is  nearly  equal  to  8  P,  so  that  we  may  take  for  an 
approximation  to  the  required  line, 

o  p  2  o  p  2  i 

SP  x  "L  =  8P  x  7Tr^£=r_^.  =  SP  x  — 


(SP+PE)*  /         PE 

=  SP(l-2^-f  etc.), 

the  last  equation  being   obtained    by  the    binomial   theorrn.     But 

PE 
the   fraction  -^-p  is  so  small,  being  less  than  ^fa,  that  its  powers 

above  the  first  will  be  small  enough  to  be  neglected.     So  we  shall 
have  for  the  required  line, 

SP-2EP. 

If,  therefore,  we  take  the  point  A  so  that  P  A  shall  be  equal  to  2 
EP,  the  attraction  of  the  sun  upon  the  earth  will  on  the  same  scale  be 
represented  by  the  line  A  S.  The  disturbing  force  which  we  seek 
is  represented  by  the  difference  between  the  attraction  of  the  sun 
upon  the  earth  and  that  of  the  same  body  upon  the  moon.  If  then 
we  suppose  the  force  A  S  to  be  applied  to  the  moon  in  the  opposite 
direction,  the  resultant  of  the  two  forces  M  8  and  S  A  will  repre- 
sent the  disturbing  force  required.  By  the  law  of  the  composition 
of  forces,  this  resultant  is  represented  by  the  line  MA. 

We  are  thus  enabled  to  construct  this  force  in  a  very  simple  man- 
ner, when  the  moon  is  in  any  given  position.  When  the  moon  is 
at  JV,  the  line  N  A  will  be  equal  to  2  E  M ;  the  disturbing  force 
will  therefore  be  represented  by  twice  the  distance  of  the  moon. 
On  the  other  hand,  when  the  moon  is  at  Q  the  three  points  E  N 
and  A  will  all  coincide.  Hence  the  disturbing  force  which  tends 
to  bring  the  moon  toward  the  earth  will  be  represented  by  the  line 
Q  E  ;  hence  the  force  which  tends  to  draw  the  moon  away  from  the 
earth  at  new  and  full  moon  is  twice  as  great  as  that  which  draws 


MOON'S  NODES.  159 

the  bodies  together  at  the  quarters.  Consequently,  upon  the  whole, 
the  tendency  of  the  sun's  attraction  is  to  dimmish  the  attraction  of 
the  earth  upon  the  moon. 

§   3.    MOTION  OP  THE  MOON'S  NODES. 

Among  the  changes  which  the  sun's  attraction  produces 
in  the  moon's  orbit,  that  which  interests  us  most  is  the 
constant  variation  in  the  plane  of  the  orbit.  This  plane 
is  indicated  by  the  path  which  the  moon  seems  to  describe 
in  its  circuit  around  the  celestial  sphere.  Simple  naked 
eye  estimates  of  the  moon's  position,  continued  during  a 
month,  would  show  that  her  path  was  always  quite  near 
the  ecliptic,  because  it  would  be  evident  to  the  eye  that, 
like  the  sun,  she  was  much  farther  north  while  passing 
from  the  vernal  to  the  autumnal  equinox  than  while  de- 
scribing the  other  half  of  her  circuit  from  the  autumnal 
to  the  vernal  equinox.  It  would  be  seen  that,  like  the 
sun,  she  was  farthest  north  in  about  six  hours  of  right  as- 
cension, and  farthest  south  when  in  about  eighteen  hours 
of  right  ascension. 

To  map  out  the  path  with  greater  precision,  we  have  to 
observe  the  position  of  the  moon  from  night  to  night  with 
a  meridian  circle.  We  thus  lay  down  her  course  among 
the  stars  in  the  same  manner  that  we  have  formerly  shown 
it  possible  to  lay  down  the  sun's  path,  or  the  ecliptic.  It 
is  thus  found  that  the  path  of  the  moon  may  be  considered 
as  a  great  circle,  making  an  angle  of  5°  with  the  ecliptic, 
and  crossing  the  ecliptic  at  this  small  angle  at  two  oppo- 
site points  of  the  heavens.  These  points  are  called  the 
moon's  nodes.  The  point  at  which  she  passes  from  the 
south  to  the  north  of  the  ecliptic  is  called  the  ascending 
node  •  that  in  which  she  passes  from  the  north  to  the 
south  is  the  descending  node.  To  illustrate  the  motion  of 
the  moon  near  the  node,  the  dotted  line  a  a  may  be  taken 
as  showing  the  path  of  the  moon,  while  the  circles  show 
her  position  at  successive  intervals  of  one  hour  as  she  is  ap- 
proaching her  ascending  node.  Position  number  9  is  exactly 


160 


ASTRONOMY. 


at   the  node.     If    we 
continue  following  her 
course  in  this  way  for 
a  week,  we  should  find 
that    she    had    moved 
about  90°,  and  attained 
her  greatest  north  lati- 
tude  at    5°    from   the 
ecliptic.     At   the   end 
of   another   week,    we 
should    find    that   she 
had  returned     to     the 
ecliptic  and  crossed  it 
at  her  descending  node. 
At  the  end  of  the  third 
week  very  nearly,  we 
should  find  that  she  had 
made  three  fourths  the 
circuit  of  the  heavens, 
and   was   now   in    her 
greatest  south  latitude, 
being  5°  south  of  the 
ecliptic.     At   the   end 
of   six   or   seven   days 
more,  we  should  again 
find  her    crossing  the 
ecliptic  at  her  ascend- 
ing node  as  before.   We 
may  thus  conceive  of 
four  cardinal  points  of 
the  moon's  orbit,  90° 
apart,  marked  by  the 
two  nodes  and  the  two 
points  of  greatest  north 
and  south  latitude. 

Motion  of  the  Nodes. 
— A  remarkable  prop- 


MOON'S  NODES.  101 

erty  of  these  points  is  that  they  are  not  fixed,  but  are  con- 
stantly moving.  The  general  motion  is  a  little  irregular, 
but,  leaving  out  small  irregularities,  it  is  constantly  toward 
the  west.  Thus  returning  to  our  watch  of  the  course  of 
the  moon,  we  should  find  that,  at  her  next  return  to  the 
ascending  node,  she  would  not  describe  the  line  a  a  as 
before,  but  the  line  J  1)  about  one  fourth  of  a  diameter 
north  of  it.  She  would  therefore  reach  the  ecliptic  more 
than  1-J0  west  of  the  preceding  point  of  crossing,  and  her 
other  cardinal  points  would  be  found  1-J0  farther  west  as 
she  went  around.  On  her  next  return  she  wrould  describe 
the  line  c  c,  then  the  line  d  d,  etc. ,  indefinitely,  each  line 
being  farther  toward  the  west.  The  figure  shows  the 
paths  in  five  consecutive  returns  to  the  node. 

A  lapse  of  nine  years  will  bring  the  descending  node 
around  to  the  place  which  was  before  occupied  by  the 
ascending  node,  and  thus  we  shall  have  the  moon  crossing 
at  a  small  inclination  toward  the  south,  as  shown  in  the 
figure. 

A  complete  revolution  of  the  nodes  takes  place  in  18-6 
years.  After  the  lapse  of  this  period,  the  motion  is  re- 
peated in  the  same  manner. 

One  consequence  of  this  motion  is  that  the  moon,  after 
leaving  a  node,  reaches  the  same  node  again  sooner  than 
she  completes  her  true  circuit  in  the  heavens.  How  much 
sooner  is  readily  computed  from  the  fact  that  the  retro- 
grade motion  of  the  node  amounts  to  1°  26'  31"  during 
the  period  that  the  moon  is  returning  to  it.  It  takes  the 
moon  about  two  hours  and  a  half  (more  exactly  Od.  10944) 
to  move  through  this  distance  ;  consequently,  comparing 
with  the  sidereal  period  already  given,  we  find  that  the 
return  of  the  moon  to  her  node  takes  place  in  27d.  32166 
—  Od.  10944  =  27d.  21222.  This  time  will  be  important  to 
us  in  considering  the  recurrence  of  eclipses. 

In  Fig.  59  is  illustrated  the  effect  of  these  changes  in 
the  position  of  the  moon's  orbit  upon  her  motion  rela- 


162 


ASTRONOMY. 


tive  to  the  equator.  E  here  represents  the  vernal  and 

A  the  autumnal  equinox,  situated 
180°  apart.  In  March,  1876, 
the  moon's  ascending  node  cor- 
responded with  the  vernal  equi- 
nox, and  her  descending  node 
with  the  autumnal  one.  Conse- 
quently she  was  5°  north  of  the 
ecliptic  when  in  six  hours  of 
right  ascension  or  near  the  mid- 
dle of  the  figure.  Since  the 
ecliptic  is  23J°  north  of  the 
equator  at  this  point,  the  moon  at- 
tained a  maximum  declination  of 
28-J0  ;  she  therefore  passed  nearer 
the  zenith  when  in  six  hours 
of  right  ascension  than  at  any 
other  time  during  the  eighteen 
years'  period.  In  the  language 
of  the  almanac,  "  the  moon  ran 
high. ' '  Of  course  when  at  her 
greatest  distance  south  of  the 
equator,  in  the  other  half  of  her 
orbit,  she  attained  a  correspond- 
ing south  declination,  and  cul- 
minated at  a  lower  altitude  than 
she  had  for  eighteen  years.  In 
1885  the  nodes  will  change  places, 
and  the  orbit  will  deviate  from 
the  equator  less  than  at  any  other 
time  during  the  eighteen  years. 
In  1880  the  descending  node  will 
be  in  six  hours  of  right  ascension, 
and  the  greatest  angular  distance 
of  the  moon  from  the  equator 

will  be  nearly  equal  to  that  of  the  sun. 


PERIGEE  OF  THE  MOON.  163 

§  4.    MOTION  OF  THE  PERIGEE. 

If  the  sun  exerted  no  disturbing  force  on  the  moon,  the 
latter  would  move  round  the  earth  in  an  ellipse  according 
to  KEPLER'S  laws.  But  the  difference  of  the  sun's  attrac- 
tion on  the  earth  and  on  the  moon,  though  only  a  small 
fraction  of  the  earth's  attractive  force  on  the  moon,  is  yet 
so  great  as  to  produce  deviations  from  the  elliptic  motion 
very  much  greater  than  occur  in  the  motions  of  the  planets. 
It  also  produces  rapid  changes  in  the  elliptic  orbit.  The 
most  remarkable  of  these  changes  are  the  progressive 
motion  of  the  nodes  just  described  and  a  corresponding 
motion  of  the  perigee.  Referring  to  Fig.  52,  which  illus- 
trated the  elliptic  orbit  of  a  planet,  let  us  suppose  it  to 
represent  the  orbit  of  the  moon.  S  will  then  represent 
the  earth  instead  of  the  sun,  and  n  will  be  the  lun^rjwr- 
iyee,  or  the  point  of  the  orbit  nearest  the  earth.  But, 
instead  of  remaining  nearly  fixed,  as  do  the  orbits  of  the 
planets,  the  lunar  orbit  itself  may  be  considered  as  making 
a  revolution  round  the  earth  in  about  nine  years,  in  the 
same  direction  as  the  moon  itself.  Hence  if  we  note  the 
longitude  of  the  moon's  perigee  at  any  time,  and  again 
two  or  three  years  later,  we  shall  find  the  two  positions 
quite  different.  If  we  wait  four  years  and  a  half,  we  shall 
find  the  perigee  in  directly  the  opposite  point  of  the 
heavens. 

The  eccentricity  of  the  moon's  orbit  is  about  0.055,  and 
in  consequence  the  moon  is  about  6°  ahead  of  its  mean 
place  when  90°  past  the  perigee,  and  about  the  same  dis- 
tance behind  when  half  way  from  apogee  to  perigee. 

The  disturbing  action  of  the  sun  produces  a  great  num- 
ber of  other  inequalities,  of  which  the  largest  are  the 
evection  and  the  variation.  The  former  is  more  than  a 
degree,  and  the  latter  not  much  less.  The  formulas  by 
which  they  are  expressed  belong  to  Celestial  Mechanics, 
and  the  reader  who  desires  to  study  them  is  referred  to 
works  on  that  subject. 


ASTRONOMY. 


§   5.    ROTATION  OP  THE  MOON. 

The  moon  rotates  on  her  axis  in  the  same  time  and  in 
the  same  direction  in  which  she  revolves  around  the  earth. 
In  consequence  she  always  presents  very  nearly  the  same 
face  to  the  earth.*  There  is  indeed  a  small  oscillation 
called  the  libration  of  the  moon,  arising  from  the  fact  that 
her  rotation  on  her  axis  is  uniform,  while  her  revolution 
around  the  earth  is  not  uniform.  In  consequence  of 
this  we  sometimes  see  a  little  of  her  farther  hemisphere 
first  on  one  side  and  then  on  the  other,  but  the  greater 
part  of  this  hemisphere  is  forever  hidden  from  human 
sight. 

The  axis  of  rotation  of  the  moon  is  inclined  to  the 
ecliptic  about  1°  29'.  It  is  remarkable  that  this  axis 
changes  its  direction  in  a  way  corresponding  exactly  to 
the  motion  of  the  nodes  of  the  moon's  orbit.  Let  us  sup- 
pose a  line  passing  through  the  centre  of  the  earth  per- 
pendicular to  the  plane  of  the  moon's  orbit.  In  conse- 
quence of  the  inclination  of  the  orbit  to  the  ecliptic,  this 
line  will  point  5°  from  the  pole  of  the  ecliptic.  Then, 
suppose  another  line  parallel  to  the  moon's  axis  of  rota- 
tion. This  line  will  intersect  the  celestial  sphere  1°  29' 
from  the  pole  of  the  ecliptic,  and  on  the  opposite  side 
from  the  pole  of  the  moon's  orbit,  so  that  it  will  be  6^° 
from  the  latter.  As  one  pole  revolves  around  the 
pole  of  the  ecliptic  in  18.6  years,  the  other  will  do  the 
same,  always  keeping  the  same  position  relative  to  the 
first. 


*  This  conclusion  is  often  a  pom  asinorum  to  some  who  conceive 
that,  if  the  same  face  of  the  moon  is  always  presented  to  the  earth,  she 
cannot  rotate  at  all.  The  difficulty  arises  from  a  misunderstanding  of 
the  difference  between  a  relative  and  an  absolute  rotation.  It  is  true 
that  she  does  not  rotate  relatively  to  the  line  drawn  from  the  earth  to 
her  centre,  but  she  must  rotate  relative  to  a  fixed  line,  or  a  line  drawn 
to  a  fixed  star. 


THE  TIDES.  105 


§  6.    THE  TIDES. 

The  ebb  and  flow  of  the  tides  are  produced  by  the  un- 
equal attraction  of  the  sun  and  moon  on  different  parts  of 
the  earth,  arising  from  the  fact  that,  owing  to  the  magni- 
tude of  the  earth,  some  parts  of  it  are  nearer  these  attracting 
bodies  than  others,  and  are  therefore  more  strongly  at- 
tracted. To  understand  the  nature  of  the  tide-producing 
force,  we  must  recall  the  principle  of  mechanics  already 
cited,  that  if  two  neighboring  bodies  are  acted  on  by 
equal  and  parallel  accelerating  forces,  their  motion  rel- 
ative to  each  other  will  not  be  altered,  because  both  will 
move  equally  under  the  influence  of  the  forces.  When 
the  forces  are  slightly  different,  either  in  magnitude  or 
direction  or  both,  the  relative  motion  of  the  two  bodies 
will  depend  on  this  difference  alone.  Since  the  sun  and 
moon  attract  those  parts  of  the  earth  which  are  nearest 
them  more  powerfully  than  those  which  are  remote,  there 
arises  an  inequality  which  produces  a  motion  in  the 
waters  of  the  ocean.  As  the  earth  revolves  on  its  axis, 
different  parts  of  it  are  brought  in  in  succession  under  the 
moon.  Thus  a  motion  is  produced  in  the  ocean  which 
goes  through  its  rise  and  fall  according  to  the  apparent 
position  of  the  moon.  This  is  called  the  tidal  wave. 

The  tide-producing  force  of  the  sun  and  moon  is  so  nearly  like 
the  disturbing  force  of  the  sun  upon  the  motion  of  the  moon  around 
the  earth  that  nearly  the  same  explanation  will  apply  to  both.  Let 
us  then  refer  again  to  Fig.  57,  and  suppose  E  to  represent  the 
centre  of  the  earth,  the  circle  F  Q  JV  its  circumference,  M  a  par- 
ticle of  water  on  the  earth's  surface,  and  8  either  the  sun  or  the 
moon. 

The  entire  earth  being  rigid,  each  part  of  it  will  move  under  the 
influence  of  the  moon's  attraction  as  if  the  whole  were  concen- 
trated at  its  centre.  But  the  attraction  of  the  moon  upon  the 
particle  M,  being  different  from  its  mean  attraction  on  the  earth,  will 
tend  to  make  it  move  differently  from  the  earth.  The  force  which 
causes  this  difference  of  motion,  as  already  explained,  will  be  repre- 
sented by  the  line  M  A.  It  is  true  that  this  same  disturbing  force  is 
acting  upon  that  portion  of  the  solid  earth  at  M  as  well  as  upon  the 
water.  But  the  earth  cannot  yield  on  account  of  its  rigidity  ;  the 


166  ASTRONOMY. 

water  therefore  tends  to  flow  along  the  earth's  surface  from  M 
toward  N.  There  is  therefore  a  residual  force  tending  to  make  the 
water  higher  at  JVthan  at  M. 

If  we  suppose  the  particle  M  to  be  near  F,  then  the  point  A  will 
be  to  the  left  of  F.  The  water  will  therefore  be  drawn  in  an  oppo- 
site direction  or  toward  F.  There  will  therefore  also  be  a  force 
tending  to  make  the  water  accumulate  around  F.  As  the  disturb- 
ing force  of  the  sun  tends  to  cause  the  earth  and  moon  to  separate 
both  at  new  and  full  moon,  so  the  tidal  force  of  the  sun  and 
moon  upon  the  earth  tends  to  make  the  waters  accumulate  both  at 
M  and  F.  More  exactly,  the  force  in  question  tends  to  draw  the 
earth  out  into  the  form  of  a  prolate  ellipsoid,  having  its  longest 
axis  in  the  direction  of  the  attracting  body.  As  the  earth  rotates 
on  its  axis,  each  particle  of  the  ocean  is,  in  the  course  of  a  day, 
brought  in  to  the  four  positions  N  Q,  F  R,  or  into  some  positions 
corresponding  to  these.  Thus,  the  tide-producing  force  changes 
back  and  forth  twice  in  the  course  of  a  lunar  day.  (By  a  lunar  day 
we  mean  the  interval  between  two  successive  passages  of  the  moon 
across  the  meridian,  which  is,  on  the  average,  about  24h  48m.)  If  the 
waters  could  yield  immediately  to  this  force,  we  should  always  have 
high  tide  at  F  and  N  and  low  tides  at  Q  and  7?.  But  there  are  two 
causes  which  prevent  this. 

1.  Owing  to  the  inertia  of  the  water,  the  force  must  act  some 
time  before  the  full  amount  of  motion  is  produced,  and  this  motion, 
once  attained,  will  continue  after  the  force    has  ceased  to  act. 
Again,   the  waters  will  continue  to  accumulate  as  Icng  as  there  is 
any  motion  in  the  required  direction.     The  result  of  this  would  be 
high  tides  at  Q  and  7?  and  low  tides  at  F  and  JVJ  if  the  ocean 
covered  the  earth  and  were  perfectly  free  to  move.     That  is,  high 
tides  would  then  be  six  hours  after  the  moon  crossed  the  meridian. 

2.  The   principal    cause,    however,  which    interferes   with   the 
regularity  of  the  motion  is  the  obstruction  of  islands  and  continents 
to  the  free  motion  of  the  water.     These  deflect  the  tidal  wave  from 
its  course  in  so  many  different  ways,  that  it  is  hardly  possible  to 
trace  the  relation  between  the  attraction  of  the  moon  and  the  mo- 
tion of  the  tide  ;  the  time  of  high  and  low  tide  must  therefore  be 
found  by  observing  at  each  point  along  the  coast.     By  comparing 
these  times  through  a  series  of  years,  a  very  accurate  idea  of  the 
motion  of  the  tidal  wave  can  be  obtained. 

Such  observations  have  been  made  over  our  Atlantic  and  Pacific 
coasts  by  the  Coast  Survey  and  over  most  of  the  coasts  of  Europe, 
by  the  countries  occupying  them.  Unfortunately  the  tides  cannot 
be  observed  away  from  the  land,  and  hence  little  is  known  of  the 
course  of  the  tidal  wave  over  the  ocean. 

We  have  remarked  that  both  the  sun  and  moon  exert  a 
tide-producing  force.  That  of  the  sun  is  about  ^  of  that 
of  the  moon.  At  new  and  full  moon  the  two  forces  are 
united,  and  the  actual  force  is  equal  to  their  sum.  At 


THE  TIDES.  167 

first  and  last  quarter,  when  the  two  bodies  are  90°  apart, 
they  act  in  opposite  directions,  the  sun  tending  to  produce 
a  high  tide  where  the  moon  tends  to  produce  a  low  one, 
and  vice  versa.  The  result  of  this  is  that  near  the  time  of 
new  and  full  moon  we  have  what  are  known  as  the  spring 
tides,  and  near  the  quarters  what  are  called  neap  tides.  If 
the  tides  were  always  proportional  to  the  force  which  pro- 
duces them,  the  spring  tides  would  be  highest  at  full 
moon,  but  the  tidal  wave  tends  to  go  on  for  some  time 
after  the  force  which  produces  it  ceases.  Hence  the  high- 
est spring  tides  are  not  reached  until  two  or  three  days  after 
new  and  full  moon.  Again,  owing  to  the  effect  of  fric- 
tion, the  neap  tides  continue  to  be  less  and  less  for  two  or 
three  days  after  the  first  and  last  quarters,  when  the  grad- 
ually increasing  force  again  has  time  to  make  itself  felt. 

The  theory  of  the  tides  offers  very  complicated  prob- 
lems, which  have  taxed  the  powers  of  mathematicians  for 
several  generations.  These  problems  are  in  their  elements 
less  simple  than  those  presented  by  the  motions  of  the 
planets,  owing  to  the  number  of  disturbing  circumstances 
which  enter  into  them.  The  various  depths  of  the  ocean 
at  different  points,  the  friction  of  the  water,  its  momen- 
tum when  it  is  once  in  motion,  the  effect  of  the  coast-lines, 
have  all  to  be  taken  into  account.  These  quantities  are 
so  far  from  being  exactly  known  that  the  theory  of  the 
tides  can  be  expressed  only  by  some  general  principles 
which  do  not  suffice  to  enable  us  to  predict  them  for  any 
given  place.  From  observation,  however,  it  is  easy  to 
construct  tables  showing  exactly  what  tides  correspond  to 
given  positions  of  the  sun  and  moon  at  any  port  where  the 
observations  are  made.  With  such  tables  the  ebb  and  flow 
are  predicted  for  the  benefit  of  all  who  are  interested,  but 
the  results  may  be  a  little  uncertain  on  account  of  the 
effect  of  the  winds  upon  the  motion  of  the  water. 


CHAPTER  VII. 

ECLIPSES   OP   THE   SUN  AND   MOON 

ECLIPSES  are  a  class  of  phenomena  arising  from  the 
shadow  of  one  body  being  cast  upon  another,  and  thus 
wholly  or  partially  obscuring  it.  In  an  eclipse  of  the  sun, 
the  shadow  of  the  moon  sweeps  over  the  earth,  and  the 
sun  is  wholly  or  partially  obscured  to  observers  on  that 
part  of  the  earth  where  the  shadow  falls.  In  an  eclipse  of 
the  moon,  the  latter  enters  the  shadow  of  the  earth,  and  is 
wholly  or  partially  obscured  in  consequence  of  being  de- 
prived of  some  or  all  its  borrowed  light.  The  satellites 
of  other  planets  are  from  time  to  time  eclipsed  in  the 
same  way  by  entering  the  shadows  of  their  primaries  ; 
among  these  the  satellites  of  Jupiter  are  objects  whose 
eclipses  may  be  observed  with  great  regularity. 

§  1.  THE  EARTH'S  SHADOW  AND  PENUMBRA. 

In  Fig.  60  let  $  represent  the  sun  and  E  the  earth. 
Draw  straight  lines,  D  B  Fand  D'  V'V,  each  tangent 
to  the  sun  and  the  earth.  The  two  bodies  being  supposed 
spherical,  these  lines  will  be  the  intersections  of  a  cone 
with  the  plane  of  the  paper,  and  may  be  taken  to  repre- 
sent that  cone.  It  is  evident  that  the  cone  E  V B'  will 
be  the  outline  of  the  shadow  of  the  earth,  and  that  within 
this  cone  no  direct  sunlight  can  penetrate.  It  is  therefore 
called  the  earth's  shadow  cone. 

Let  us  also  draw  the  lines  D'  B  P  and  D  B'  P'  to  rep- 
resent the  other  cone  tangent  to  the  sun  and  earth.  It  is 


THE  EARTH'S  SHADOW. 


169 


then  evident  that  within  the  region  V B  P  and  V  B'  P' 
the  light  of  the  sun  will  be  partially  but  not  entirely  cut 
oif. 


FlG.  60. — FORM  OF   SHADOW. 

Dimensions  of  Shadow.  —Let  us  investigate  the  distance  E  Ffrom 
the  centre  of  the  earth  to  the  vertex  of  the  shadow.  The  triangles 
V  E  B  and  VS  D  are  similar,  having  a  right  angle  at  B  and  at  D. 
Hence, 

V  E:  EB  =  VS:  SD  =  ES:  (SD-EB). 
So  if  we  put 

l=VE,  the  length  of  the  shadow  measured  from  the^  centre  of 
the  earth. 

r  =  ES,  the  radius  vector  of  the  earth, 
R  =  S  D,  the  radius  of  the  sun, 
p  =  E  B,  the  radius  of  the  earth, 

$  the  angular  serni-diameter  of  the  sun  as  seen  from  the  earth, 
TT,  the  horizontal  parallax  of  the  sun, 

we  have 


ES  x  EB 


rp 


But  by  the  theory  of  parallaxes  (Chapter  I.,  §  7), 


p  =  r  sin  TT 
R  =  r  sin  8 


Hence, 


sin  S  —  sin  TT" 


The  mean  value  of  the  sun's  angular  semi-diameter,  from  which 
the  real  value  never  differs  by  more  than  the  sixtieth  part,  is  found 
by  observations  to  be  about  16'  0"  =  960",  while  the  mean  value  of  TT 


170  ASTRONOMY. 

is  about  8"  •  8.    We  find  sin  8—  sin  TT  =  0  •  00461,  and  -r— : = 

sm  8  —  sin  •* 

.__i_^T  =  217.  We  therefore  conclude  that  the  mean  length  of 
the  earth's  shadow  is  217  times  the  earth's  radius  ;  in  round 
numbers  1,380,000  kilometres,  or  800,000  miles,  the  mean  radius 
of  the  earth  being  6370  kilometres.  It  will  be  seen  from  the  figure 
that  it  varies  directly  as  the  distance  of  the  earth  from  the 
sun  ;  it  is  therefore  about  one  sixtieth  less  than  the  mean  in  Decem- 
ber, and  one  sixtieth  greater  in  June. 

The  radius  of  the  shadow  diminishes  uniformly  with  the  distance 
as  we  go  outward  from  the  earth.     At  any  distance   z  from  the 

earth's  centre  it  will  be  equal  to  f  1  —  ^  Jp,  for  this  formula  gives 

the  radius  p  when  2  =  0,  and  the  diameter  zero  when  z  —  I  as  it 
should.* 


§  2.    ECLIPSES  OP  THE  MOON. 

The  mean  distance  of  the  moon  from  the  earth  is  about 
60  radii  of  the  latter,  while,  as  we  have  just  seen,  the 
length  E  V  of  the  earth's  shadow  is  217  radii  of  the  earth. 
Hence  when  the  moon  passes  through  the  shadow  she  does 
so  at  a  point  less  than  three  tenths  of  the  way  from 
E  to  F  *  The  radius  of  the  shadow  here  will  be  *V-rr^ 
of  the  radius  E  B  of  the  earth,  a  quantity  which  we  read- 
ily find  to  be  about  4600  kilometres.  The  radius  of  the 
moon  being  1736  kilometres,  it  will  be  entirely  enveloped 
by  the  shadow  when  it  passes  through  it  within  2864 
kilometres  of  the  axis  E  V  of  the  shadow.  If  its  least  dis- 
tance from  the  axis  exceed  this  amount,  a  portion  of  the 
lunar  globe  will  be  outside  the  limits  B  V  of  the  shadow 
cone,  and  will  therefore  receive  a  portion  of  the  direct 
light  of  the  sun.  If  the  least  distance  of  the  centre  of  the 
moon  from  the  axis  of  the  shadow  is  greater  than  the 
sum  of  the  radii  of  the  moon  and  the  shadow — that  is, 
greater  than  6336  kilometres — the  moon  will  not  enter  the 

*  It  will  be  noted  that  this  expression  is  not,  rigorously  speaking,  the 
semi-diameter  of  the  shadow,  but  the  shortest  distance  from  a  point  on 
its  central  line  to  its  conical  surface.  This  distance  is  measured  in  a 
direction  E  B  perpendicular  to  D  B,  whereas  the  diameter  would  be 
perpendicular  to  the  axis  8  E,  and  its  half  length  would  be  a  little 
greater  than  E  B. 


ECLIPSES  OF  THE  MOON.  171 

shadow  at  all,  and  there  will  he  no  eclipse  proper,  though 
the  brilliancy  of  the  moon  must  he  diminished  wherever 
she  is  within  the  penumbra!  region. 

When  an  eclipse  of  the  moon  occurs,  the  phases  are  laid 
down  in  the  almanac  in  the  following  manner  :  Supposing 
the  moon  to  he  moving  around  the  earth  from  below  up- 
ward, its  advancing  edge  first  meets  the  boundary  1?  Pf 
of  the  penumbra.  The  time  of  this  occurrence  is  given  in 
the  almanac  as  that  of  "  moon  entering  penumbra."  A 
small  portion  of  the  sunlight  is  then  cut  off  from  the  ad- 
vancing edge  of  the  moon,  and  this  amount  constantly  in- 
creases until  the  edge  reaches  the  boundary  Br  V  of  the 
shadow.  It  is  curious,  however,  that  the  eye  can  scarcely 
detect  any  diminution  in  the  brilliancy  of  the  moon  until 
she  has  almost  touched  the  boundary  of  the  shadow.  The 
observer  must  not  therefore  expect  to  detect  the  coming 
eclipse  until  very  nearly  the  time  given  in  the  almanac  as 
that  of  "  moon  entering  shadow."  As  this  happens,  the 
advancing  portion  of  the  lunar  disk  will  be  entirely  lost  to 
view,  as  if  it  were  cut  off  hy  a  rather  ill-defined  line.  It 
takes  the  moon  about  an  hour  to  move  over  a  distance 
equal  to  her  own  diameter,  so  that  if  the  eclipse  is  nearly 
central  the  whole  moon  will  be  immersed  in  the  shadow 
about  an  hour  after  she  first  strikes  it.  This  is  the  time  of 
beginning  of  total  eclipse.  So  long  as  only  a  moderate 
portion  of  the  moon's  disk  is  in  the  shadow,  that  portion 
will  be  entirely  invisible,  but  if  the  eclipse  becomes  total 
the  whole  disk  of  the  moon  will  nearly  always  be  plainly 
visible,  shining  with  a  red  coppery  light.  This  is  owing  to 
the  refraction  of  the  sun's  rays  by  the  lower  strata  of  the 
earth's  atmosphere.  We  shall  see  hereafter  that  if  a  ray  of 
light  D  B  passes  from  the  sun  to  the  earth,  so  as  just  to 
graze  the  latter,  it  is  bent  by  refraction  more  than  a  de- 
gree out  of  its  course,  so  that  at  the  distance  of  the  moon 
the  whole  shadow  is  filled  with  this  refracted  light.  An 
observer  on  the  moon  would,  during  a  total  eclipse  of  the 
latter,  see  the  earth  surrounded  by  a  ring  of  light,  and  this 


172  ASTRONOMY. 

ring  would  appear  red,  owing  to  the  absorption  of  the  blue 
and  green  rays  by  the  earth's  atmosphere,  just  as  the  sun 
seems  red  when  setting. 

The  moon  may  remain  enveloped  in  the  shadow  of  the 
earth  during  a  period  ranging  from  a  few  minutes  to  nearly 
two  hours,  according  to  the  distance  at  which  she  passes 
from  the  axis  of  the  shadow  and  the  velocity  of  her  angu- 
lar motion.  When  she  leaves  the  shadow,  the  phases 
which  we  have  described  occur  in  reverse  order. 

It  very  often  happens  that  the  moon  passes  through  the 
penumbra  of  the  earth  without  touching  the  shadow  at  all. 
No  notice  is  taken  of  these  passages  in  our  almanacs,  be- 
cause, as  already  stated,  the  diminution  of  light  is  scarcely 
perceptible  unless  the  moon  at  least  grazes  the  edge  of  the 
shadow. 

§  3.    ECLIPSES  OF  THE  SUN. 

In  Fig.  57  we  may  suppose  B  E  B'  to  represent  the 
moon  as  well  as  the  earth.  The  geometrical  theory  of  the 
shadow  will  remain  the  same,  though  the  length  of  the 
shadow  will  be  much  less.  We  may  regard  the  mean 
semi-diameter  of  the  sun  as  seen  from  the  moon,  and  its 
mean  parallax,  as  being  the  same  for  the  moon  as  for  the 
earth.  Therefore  in  the  formula  which  gives  the  length 
of  the  moon's  shadow  the  denominator  will  retain  the 
same  value,  while  in  the  numerator  we  must  substitute  the 
radius  of  the  moon  for  that  of  the  earth.  The  radius  of 
the  moon  is  about  1736  kilometres,  or  1080  miles.  Multi- 
plying this  by  217,  as  before,  we  find  the  mean  length  of 
the  moon's  shadow  to  be  377,000  kilometres,  or  235,000 
miles.  This  is  very  nearly  the  same  with  the  distance  of 
the  moon  from  the  earth  when  she  is  in  conjunction  with 
the  sun.  We  therefore  conclude  that  when  the  moon 
passes  between  the  earth  and  the  sun,  the  former  will  be 
very  near  the  vertex  V  of  the  shadow.  As  a  matter  of 
fact,  an  observer  on  the  earth's  surface  will  sometimes  pass 


THE  MOON'S  SHADOW.  173 

through  the  region   G  V  C,  and  sometimes  on  the  other 
side  of  F. 

Now,  in  Fig.  60,  still  supposing  B  E  B'  to  be  the 
moon,  let  us  draw  the  lines  D  HP*  and  D'  B  P  tan- 
gent to  both  the  moon  and  the  sun,  but  crossing  each  other 
between  these  bodies  at  I.  It  is  evident  that  outside  the 
space  P  B  B'  P'  an  observer  will  see  the  whole  sun,  no 
part  of  the  moon  being  projected  upon  it  ;  while  within 
this  space  the  sun  will  be  more  or  less  obscured.  The 
whole  obscured  space  may  be  divided  into  three  regions,  in 
each  of  which  the  character  of  the  phenomenon  is  differ- 
ent from  what  it  is  in  the  others. 

Firstly,  we  have  the  region  B  V B'  forming  the  shadow 
cone  proper.  Here  the  sunlight  is  entirely  cut  off  by  the 
moon,  and  darkness  is  therefore  complete,  except  so  far  as 
light  may  enter  by  refraction  or  reflection.  To  an  observer 
at  V  the  moon  would  exactly  cover  the  sun,  the  two 
bodies  being  apparently  tangent  to  each  other  all  around. 

Secondly,  we  have  the  conical  region  to  the  right  of  V 
between  the  lines  B  T^and  B'  V  continued.  In  this 
region  the  moon  is  seen  wholly  projected  upon  the  sun, 
the  visible  portion  of  the  latter  presenting  the  form  of  a 
ring  of  light  around  the  moon.  This  ring  of  light  will  be 
wider  in  proportion  to  the  apparent  diameter  of  the  sun, 
the  farther  out  we  go,  because  the  moon  will  appear 
smaller  than  the  sun,  and  its  angular  diameter  will  dimin- 
ish in  a  more  rapid  ratio  than  that  of  the  sun.  This 
region  is  that  of  annular  eclipse,  because  the  sun  will  pre- 
sent the  appearance  of  an  annulus  or  ring  of  light  around 
the  moon. 

Thirdly,  we  have  the  region  P  B  V  and  P'  B'  V,  which 
we  notice  is  connected,  extending  around  the  interior  cone. 
An  observer  here  would  see  the  moon  partly  projected 
upon  the  sun,  and  therefore  a  certain  part  of  the  sun's 
light  would  be  cut  off.  Along  the  inner  boundary  B  V 
and  B'  V  the  obscuration  of  the  sun  will  be  complete, 
but  the  amount  of  sunlight  will  gradually  increase  out  to 


174 


ASTRONOMY. 


the  outer  boundary  B  P  B'  Pr ,  where  the  whole  sun  is 
visible.  This  region  of  partial  obscuration  is  called  the 
penumbra. 

To  show  more  clearly  the  phenomena  of  solar  eclipse, 
we  present  another  figure  representing  the  penumbra  of 


FlG.    61. — FIGURE  OF  SHADOW  FOK  ANNULAR   ECLIPSE. 

the  moon  thrown  upon  the  earth.*  The  outer  of  the  two 
circles  8  represents  the  limb  of  the  sun.  The  exterior  tan- 
gents which  mark  the  boundary  of  the  shadow  cross  each 
other  at  V  before  reaching  the  earth.  The  earth  being 
a  little  beyond  the  vertex  of  the  shadow,  there  can  be  no 
total  eclipse.  In  this  case  an  observer  in  the  penumbral 
region,  C  0  or  D  #,  will  see  the  moon  partly  projected  on 
the  sun,  while  if  he  chance  to  be  situated  at  O  he  will  see 
an  annular  eclipse.  To  show  how  this  is,  we  draw  dotted 
lines  from  0  tangent  to  the  moon.  The  angle  between 
these  lines  represents  the  apparent  diameter  of  the  moon 
as  seen  from  the  earth.  Continuing  them  to  the  sun,  they 
show  the  apparent  diameter  of  the  moon  as  projected  upon 
the  sun.  It  will  be  seen  that  in  the  case  supposed,  when 

*  It  will  be  noted  that  all  the  figures  of  eclipses  are  necessarily  drawn 
very  much  out  of  proportion.  Really  the  sun  is  400  times  the  distance 
of  the  moon,  which  again  is  60  times  the  radius  of  the  earth.  But  it 
would  be  entirely  impossible  to  draw  a  figure  of  this  proportion  ;  we 
are  therefore  obliged  to  represent  the  earth  as  larger  than  the  sun,  and 
the  moon  as  nearly  half  way  between  the  earth  and  sun. 


ECLIPSES  OF  THE  SUN.  175 

the  vertex  of  the  shadow  is  between  the  earth  and  moon, 
the  latter  will  necessarily  appear  smaller  than  the  sun,  and 
the  observer  will  see  a  portion  of  the  solar  disk  on  all 
sides  of  the  moon,  as  shown  in  Fig.  62. 

If  the  moon  were  a  little  nearer  the  earth  than  it  is  rep- 
resented in  the  figure,  its  shadow  would  reach  the  earth 


FlG.  62. — DARK    BODY    OF   MOON  PROJECTED  ON  SUN  DURING   AN 
ANNULAR  ECLIPSE. 

in  the  neighborhood  of  0.  We  should  then  have  a  total 
eclipse  at  each  point  of  the  earth  on  which  it  fell.  It  will 
be  seen,  however,  that  a  total  or  annular  eclipse  of  the  sun 
is  visible  only  on  a  very  small  portion  of  the  earth's  sur- 
face, because  the  distance  of  the  moon  changes  so  little 
that  the  earth  can  never  be  far  from  the  vertex  V  of  the 
shadow.  As  the  moon  moves  around  the  earth  from  west 
to  east,  its  shadow,  whether  the  eclipse  be  total  or  annu- 
lar, moves  in  the  same  direction.  The  diameter  of  the 
shadow  at  the  surface  of  the  earth  ranges  from  zero  to  150 
miles.  It  therefore  sweeps  along  a  belt  of  the  earth's  sur- 
face of  that  breadth,  in  the  same  direction  in  which  the 
earth  is  rotating.  The  velocity  of  the  moon  relative  to 
the  earth  being  3400  kilometres  per  hour,  the  shadow 
would  pass  along  with  this  velocity  if  the  earth  did  not  ro- 
tate, but  owing  to  the  earth's  rotation  the  velocity  relative 


176  ASTRONOMY. 

to  points  on  its  surface  may  range  from  2000  to  3400 
kilometres  (1200  to  2100  miles). 

The  reader  will  readily  understand  that  in  order  to  see 
a  total  eclipse  an  observer  must  station  himself  before- 
hand at  some  point  of  the  earth's  surface  over  which  the 
shadow  is  to  pass.  These  points  are  generally  calculated 
some  years  in  advance,  in  the  astronomical  ephemerides, 
with  as  much  precision  as^  the  tables  of  the  celestial  mo- 
tions admit  of. 

It  will  be  seen  that  a  partial  eclipse  of  the  sun  may  be 
visible  from  a  much  larger  portion  of  the  earth's  surface 
than  a  total  or  annular  one.  The  space  CD  (Fig.  61)  over 
which  the  penumbra  extends  is  generally  of  about  one  half 
the  diameter  of  the  earth.  Roughly  speaking,  a  partial 
eclipse  of  the  sun  may  sweep  over  a  portion  of  the  earth's 
surface  ranging  from  zero  to  perhaps  one  fifth  or  one  sixth 
of  the  whole. 

There  are  really  more  eclipses  of  the  sun  than  of  the 
moon.  A  year  never  passes  without  at  least  two  of  the 
former,  and  sometimes  five  or  six,  while  there  are  rarely 
more  than  two  eclipses  of  the  moon,  and  in  many  years 
none  at  all.  But  at  any  one  place  more  eclipses  of  the  moon 
will  be  seen  than  of  the  sun.  The  reason  of  this  is  that 
an  eclipse  of  the  moon  is  visible  over  the  entire  hemi- 
sphere of  the  earth  on  which  the  moon  is  shining,  and  as  it 
lasts  several  hours,  observers  who  are  not  in  this  hemi- 
sphere at  the  beginning  of  the  eclipse  may,  by  the  earth's  ro- 
tation, be  brought  into  it  before  it  ends.  Thus  the  eclipse 
will  be  seen  over  more  than  half  the  earth's  surface.  But, 
as  we  have  just  seen,  each  eclipse  of  the  sun  can  be  seen 
over  only  so  small  a  fraction  of  the  earth's  surface  as  to 
more  than  compensate  for  the  greater  absolute  frequency 
of  solar  eclipses. 

It  will  be  seen  that  in  order  to  have  either  a  total  or  an- 
nular eclipse  visible  upon  the  earth,  the  line  joining  the 
centres  of  the  sun  and  moon,  being  continued,  must 
strike  the  earth.  To  an  observer  on  this  line,  the  centres 


RECUItHENCE  OF  ECLIPSES.  177 

of  the  two  bodies  will  seem  to  coincide.  An  eclipse  in 
which  this  occurs  is  called  a  central  one,  whether  it  be 
total  or  annular.  The  accompanying  figure  will  perhaps 
aid  in  giving  a  clear  idea  of  the  phenomena  of  eclipses  of 
both  sun  and  moon. 


FlG.  63. — COMPARISON  OP  SHADOW  AND  PENUMBRA  OF  EARTH  AND 
MOON.  A  IS  THE  POSITION  OF  THE  MOON  DURING  A  SOLAR,  B  DUR- 
ING A  LUNAR  ECLIPSE. 

§   4.    THE  RECURRENCE  OP  ECLIPSES. 

If  the  orbit  of  the  moon  around  the  earth  were  in  or 
near  the  same  plane  with  that  of  the  latter  around  the  sun 
—that  is,  in  or  near  the  plane  of  the  ecliptic — it  will  be 
readily  seen  that  there  would  be  an  eclipse  of  the  sun  at 
every  new  moon,  and  an  eclipse  of  the  moon  at  every 
full  moon.  But  owing  to  the  inclination  of  the  moon's 
orbit,  described  in  the  last  chapter,  the  shadow  and 
penumbra  of  the  moon  commonly  pass  above  or  below  the 
earth  at  the  time  of  new  moon,  while  the  moon,  at  her 
full,  commonly  passes  above  or  below  the  shadow  of  the 
earth.  It  is  only  when  at  the  moment  of  new  or  full'  moon 
the  moon  is  near  its  node  that  an  eclipse  can  occur. 

The  question  now  arises,  how  near  must  the  moon  be  to 
its  node  in  order  that  an  eclipse  may  occur  ?  It  is  found 
by  a  trigonometrical  computation  that  if,  at  the  moment 
of  new  moon,  the  moon  is  more  than  18° -6  from  its 
node,  no  eclipse  of  the  sun  is  possible,  while  if  it  is  less 
than  13°  •  Y  an  eclipse  is  certain.  Between  these  limits  an 
eclipse  may  occur  or  fail  according  to  the  respective  dis- 
tances of  the  sun  and  moon  from  the  earth.  Ilalf  way  be- 
tween these  limits,  or  say  16°  from  the  node,  it  is  an  even 


178  ASTRONOMY. 

chance  that  an  eclipse  will  occur  ;  toward  the  lower  limit 
(13° -7)  the  .chances  increase  to  certainty;  toward  the 
upper  one  (18° -6)  they  diminish  to  zero.  The  correspond- 
ing limits  for  an  eclipse  of  the  moon  are  9°  and  12£°—  that 
is,  if  at  the  moment  of  full  moon  the  distance  of  the 
moon  from  her  node  is  greater  than  12J°  no  eclipse  can 
occur,  while  if  the  distance  is  less  than  9°  an  eclipse  is  cer- 
tain. We  may  put  the  mean  limit  at  11°.  Since,  in  the 
long  run,  new  and  full  moon  will  occur  equally  at  all  dis- 
tances from  the  node,  there  will  be,  on  the  average,  sixteen 
eclipses  of  the  sun  to  eleven  of  the  moon,  or  nearly  fifty  per 
cent  more. 


FIG.  64.— Illustrating  lunar  eclipse  at  different  distances  from  the  node.  The  dark 
circles  are  the  earth's  shadow,  the  centre  of  which  is  always  in  the  ecliptic  A  B.  The 
moon's  orbit  is  represented  by  CD.  At  G  the  eclipse  is  central  and  total,  at  Fit  is 
partial,  and  at  E  there  is  barely  an  eclipse. 

As  an  illustration  of  these  computations,  let  us  investigate  the  lim- 
its within  which  a  central  eclipse  of  the  sun,  total  or  annular,  can 
occur.  To  allow  of  such  an  eclipse,  it  is  evident,  from  an  inspec- 
tion of  Fig.  61  or  63  that  the  actual  distance  of  the  moon  from 
the  plane  of  the  ecliptic  must  be  less  than  the  earth's  radius, 
because  the  line  joining  the  centres  of  the  sun  and  earth  always  lies 
in  this  plane.  This  distance  must,  therefore,  be  less  than  6370  kilo- 
metres. The  mean  distance  of  the  moon  being  384,000  kilometres, 
the  sine  of  the  latitude  at  this  limit  is  7ff£ro,  and  the  latitude  itself 
is  57'.  The  formula  for  the  latitude  is,  by  spherical  trigonometry, 

sin  latitude  =  sin  i  sin  u, 

i  being  the  inclination  of  the  moon's  orbit  (5°  8'),  and  uthe  distance 
of  the  moon  from  the  node.  The  value  of  sin  i  is  not  far  from  -^y, 
while,  in  a  rough  calculation,  we  may  suppose  the  comparatively 
small  angles  u  and  the  latitude  to  be  the  same  as  their  sines.  We 
may,  therefore,  suppose 

u  =  11  latitude  =  10|°. 


RECURRENCE  OF  ECLIPSES.  179 

We  therefore  conclude  that  if,  at  the  moment  of  new  moon,  the 
distance  of  the  moon  from  the  node  is  less  than  10|°  there  will  be 
a  central  eclipse  of  the  sun,  and  if  greater  than  this  there  will  not  be 
such  an  eclipse.  The  eclipse  limit  may  range  half  a  degree  or  more 
on  each  side  of  this  mean  value,  owing  to  the  varying  distance  of 
the  moon  from  the  earth.  Inside  of  10J  a  central  eclipse  may  be  re- 
garded as  certain,  and  outside  of  11°  as  impossible. 


If  the  direction  of  the  moon's  nodes  from  the  centre  of 
the  earth  were  invariable,  eclipses  could  occur  only  at  the 
two  opposite  months  of  the  year  when  the  sun  had  nearly 
the  same  longitude  as  one  node.  For  instance,  if  the  lon- 
gitudes of  the  two  opposite  nodes  were  respectively  54° 
and  234°,  then,  since  the  sun  must  be  within  12°  of  the 
node  to  allow  of  an  eclipse  of  the  moon,  its  longitude 
would  have  to  be  either  between  42°  and  66°,  or  between 
222°  and  246°.  But  the  sun  is  within  the  first  of  these  re- 
gions only  in  the  month  of  May,  and  within  the  second  only 
during  the  month  of  November.  Hence  lunar  eclipses 
could  then  occur  only  during  the  months  of  May  and  No- 
vember, and  the  same  would  hold  true  of  central  eclipses 
of  the  sun.  Small  partial  eclipses  of  the  latter  might  be 
seen  occasionally  a  day  or  two  from  the  beginnings  or  ends 
of  the  above  months,  but  they  would  be  very  small  and 
quite  rare.  Now,  the  nodes  of  the  moon's  orbit  were  act- 
ually in  the  above  directions  in  the  year  1873.  Hence 
during  that  year  eclipses  occurred  only  in  May  and  No- 
vember. We  may  call  these  months  the  seasons  of  eclipses 
for  1873. 

But  it  was  explained  in  the  last  chapter  that  there  is  a 
retrograde  motion  of  the  moon's  nodes  amounting  to  19^° 
in  a  year.  The  nodes  thus  move  back  to  meet  the  sun  in 
its  annual  revolution,  and  this  meeting  occurs  about  20  days 
earlier  every  year  than  it  did  the  year  before.  The  re- 
sult is  that  the  season  of  eclipses  is  constantly  shifting,  so 
that  each  season  ranges  throughout  the  whole  year  in  18-6 
years.  For  instance,  the  season  corresponding  to  that  of 
November,  1873,  had  moved  back  to  July  and  August  in 


180  ASTRONOMY. 

1878,  and  will  occur  in  May,  1882,  while  that  of  May, 
1873,  will  be  shifting  back  to  November  in  1882. 

It  may  be  interesting  to  illustrate  this  by  giving  the 
clays  in  which  the  sun  is  in  conjunction  with  the  nodes  of 
the  moon's  orbit  during  several  years. 

Asceuding  Node.  Descending  Node. 

1879.  January  24.  1879.   July  17. 

1880.  January  6.  1880.  June  27. 

1880.  December  18.  1881.   June    8. 

1881.  November  30.  1882.  May  20. 

1882.  November  12.  1883.  May    1. 

1883.  October  25.  1884.   April  12. 

1884.  Octobers.  1885.  March  25. 

During  these  years,  eclipses  of  the  moon  can  occur  only 
within  11  or  12  days  of  these  dates,  and  eclipses  of  the 
sun  only  within  15  or  16  days. 

In  consequence  of  the  motion  of  the  moon's  node,  three 
varying  angles  come  into  play  in  considering  the  occur- 
rence of  an  eclipse,  the  longitude  of  the  node,  that  of  tho 
sun,  and  that  of  the  moon.  We  may,  however,  simplify 
the  matter  by  referring  the  directions  of  the  sun  and 
moon,  not  to  any  fixed  line,  but  to  the  node — that  is,  we 
may  count  the  longitudes  of  these  bodies  from  the  node 
instead  of  from  the  vernal  equinox.  We  have  seen  in  the 
last  chapter  that  one  revolution  of  the  inoon  relatively  to 
the  node  is  accomplished,  on  the  average,  in  27-21222 
days.  If  we  calculate  the  time  required  for  the  sun  to  re- 
turn to  the  node,  we  shall  find  it  to  be  346  •  6201  days. 

Now,  let  us  suppose  the  sun  and  moon  to  start  out 
together  from  a  node.  At  the  end  of  346  -  6201  days  the 
sun,  having  apparently  performed  nearly  an  entire  rev- 
olution around  the  celestial  sphere,  will  again  be  at  the 
same  node,  which  has  moved  back  to  meet  it.  But  the 
moon  will  not  be  there.  It  will,  during  the  interval,  have 
passed  the  node  12  times,  and  the  13th  passage  will  not 
occur  for  a  week.  The  same  thing  will  be  true  for 


RECURRENCE  OF  ECLIPSES.  181 

18  successive  returns  of  the  sun  to  the  node  ;  we  shall 
riot  h'nd  the  moon  there  at  the  same  time  with  the  sun  ; 
she  will  always  have  passed  a  little  sooner  or  a  little  later. 
But  at  the  19th  return  of  the  sun  and  the  242d  of  the 
moon,  the  two  bodies  will  be  in  conjunction  within  half 
a  degree  of  the  node.  We  find  from  the  preceding 
periods  that 

242  returns  of  the  moon  to  the  node  require  6585  •  357  days. 
19       "         "      sun     "         "         "  6585-780    " 

The  two  bodies  will  therefore  pass  the  node  within  10 
hours  of  each  other.  This  conjunction  of  the  sun  and 
moon  will  be  the  223d  new  moon  after  that  from  which 
we  started.  Now,  one  lunation  (that  is,  the  interval 
between  two  consecutive  new  moons)  is,  in  the  mean, 
29-530588  days  ;  223  lunations  therefore  require  6585-32 
days.  The  new  moon,  therefore,  occurs  a  little  before  the 
bodies  reach  the  node,  the  distance  from  the  latter  being 
that  over  which  the  moon  moves  in  Od  •  036,  or  the  sun  in 
Od-459.  We  readily  find  this  distance  to  be  28' of  arc, 
somewhat  less  than  the  apparent  semidiameter  of  either 
body.  This  would  be  the  smallest  distance  from  either 
node  at  which  any  new  moon  would  occur  during  the 
whole  period.  The  next  nearest  approaches  would  have 
occurred  at  the  35th  and  47th  lunations  respectively. 
The  35th  new  moon  would  have  occurred  about  6°  before 
the  two  bodies  arrived  at  the  node  from  which  we  started, 
arid  the  47th  about  1  £°  past  the  opposite  node.  No  other 
new  moon  would  occur  so  near  a  node  before  the  223d 
one,  which,  as  we  have  just  seen,  would  occur  0°  28' 
west  of  the  node.  This  period  of  223  new  moons,  or  18 
years  11  days,  was  called  the  Saros  by  the  ancient  astron- 
omers. 

It  will  be  seen  that  in  the  preceding  calculations  we  have  assumed 
the  sun  and  moon  to  move  uniformly,  so  that  the  successive  new 
moon's  occurred  at  equal  intervals  of  29-530588  days,  and  at  equal 
angular  distances  around  the  ecliptic.  In  fact,  however,  the  month- 
ly inequalities  in  the  motion  of  the  moon  cause  deviations  from  her 


182  ASTRONOMY. 

mean  motion  which  amount  to  six  degrees  in  either  direction,  while 
the  annual  inequality  in  the  motion  of  the  sun  in  longitude  is  nearly 
two  degrees.  Consequently,  our  conclusions  respecting  the  point  at 
which  new  moon  occurs  may  be  astray  by  eight  degrees,  owing  to 
these  inequalities. 

But  there  is  a  remarkable  feature  connected  with  the  Saros  which 
greatly  reduces  these  inequalities.  It  is  that  this  period  of  6585£ 
days  corresponds  very  nearly  to  an  integral  number  of  revolutions 
both  of  the  earth  round  the  sun,  and  of  the  lunar  perigee  around 
the  earth.  Hence  the  inequalities  both  of  the  moon  and  of  the 
sun  will  be  nearly  the  same  at  the  beginning  and  the  end  of  a  Saros. 
In  fact,  6585^  days  is  about  18  years  and  11  days,  in  which  time 
the  earth  will  have  made  18  revolutions,  and  about  11°  on  the 
19th  revolution.  The  longitude  of  the  sun  will  therefore  be  about 
11°  greater  than  at  the  beginning  of  the  period.  Again,  in  the 
same  period  the  moon's  perigee  will  have  made  two  revolutions, 
and  will  have  advanced  13°  38'  on  the  third  revolution.  The  sun 
and  moon  being  11°  further  advanced  in  longitude,  the  conjunction 
will  fall  at  the  same  distance  from  the  lunar  perigee  within  two  or 
three  degrees.  Without  going  through  the  details  of  the  calcula- 
tion, we  may  say  as  the  result  of  this  remarkable  coincidence  that 
the  time  of  the  223d  lunation  will  not  generally  be  accelerated  or 
retarded  more  than  half  an  hour,  though  those  of  the  intermediate 
lunations  will  sometimes  deviate  more  than  half  a  day.  Also  that 
the  distance  west  of  the  node  at  which  the  new  moon  occurs  will 
not  generally  differ  from  its  mean  value,  28'  by  more  than  20'. 

In  the  preceding  explanation,  we  have  supposed  the  sun 
and  moon  to  start  out  together  from  one  of  the  nodes  of 
the  moon's  orbit.  It  is  evident,  however,  that  we  might 
have  supposed  them  to  start  from  any  given  distance  east 
or  west  of  the  node,  and  should  then  at  the  end  of  the  223d 
lunation  find  them  together  again  at  nearly  that  distance 
from  the  node.  For  instance,  on  the  5th  day  of  May, 
1864,  at  seven  o'clock  in  the  evening,  Washington  time, 
new  moon  occurred  with  the  sun  and  moon  2°  25'  west  of 
the  descending  node  of  the  moon's  orbit.  Counting  for- 
ward 223  lunations,  we  arrive  at  the  16th  day  of  May, 
1882,  when  we  find  the  new  moon  to  occur  3°  20'  west  of 
the  same  node.  Since  the  character  of  the  eclipse  depends 
principally  upon  the  relative  position  of  the  sun,  the  moon, 
and  the  node,  the  result  to  which  we  are  led  may  be  stated 
as  follows  : 

Let  us  note  the  time  of  the   middle  of  any  eclipse, 


RECURRENCE  OF  ECLIPSES.  183 

whether  of  the  sun  or  of  the  moon.  Then  let  us  go  for- 
ward 6585  days,  7  hours,  42  minutes,  and  we  shall  find 
another  eclipse  very  similar  to  the  first.  Reduced  to  years, 
the  interval  will  be  18  years  and  10  or  11  days,  according 
as  a  29th  day  of  February  intervenes  four  or  five  times 
during  the  interval.  This  being  true  of  every  eclipse,  it 
follows  that  if  we  record  all  the  eclipses  which  occur  dur- 
ing a  period  of  18  years,  we  shall  find  a  new  set  to  begin 
over  again.  If  the  period  were  an  integral  number  of 
days,  each  eclipse  of  the  new  set  would  be  visible  in  the 
same  regions  of  the  earth  as  the  old  one,  but  since  there  is 
a  fraction  of  nearly  8  hours  over  the  round  number  of 
days,  the  earth  will  be  one  third  of  a  revolution  further 
advanced  before  any  eclipse  of  the  new  set  begins.  Each 
eclipse  of  the  new  set  will  therefore  occur  about  one  third 
of  the  way  round  the  world,  or  120°  in  longitude  west  of 
the  region  in  which  the  old  one  occurred.  The  recur- 
rence will  not  take  place  near  the  same  region  until  the  end 
of  three  periods,  or  54  years  ;  and  then,  since  there  is  a 
slight  deviation  in  the  series,  owing  to  each  new  or  full 
moon  occurring  a  little  further  west  from  the  node,  the 
fourth  eclipse,  though  near  the  same  region,  will  not 
necessarily  be  similar  in  all  its  particiilars.  For  example, 
if  it  be  a  total  eclipse  of  the  sun,  the  path  of  the  shadow 
may  be  a  thousand  miles  distant  from  the  path  of  54  years 
previously. 

As  a  recent  example  of  the  Saros,  we  may  cite  some 
total  eclipses  of  the  sun  well  known  in  recent  times  ;  for 
instance  : 

1842,  July  8th,  lh  A.M.,  total  eclipse  observed  in 
Europe  ; 

1860,  July  18th,  9h  A.M.,  total  eclipse  in  America  and 
Spain  ; 

1878,  July  29th,  4h  P.M.,  one  visible  in  Texas,  Col- 
orado, and  on  the  coast  of  Alaska. 

A  yet  more  remarkable  series  of  total  eclipses  of  the 


184  ASTRONOMY. 

sun  are  those  of  the  years  1850,  1868,  1886,  etc.,  the  dates 
and  regions  being : 

1850,  August  7th,  41'  P.M.,  in  the  Pacific  Ocean  ; 

1868,  August  17th,  12h  P.M.,  in  India  ; 

1886,  August  29th,  8'1  A.M.,  in  the  Central  Atlantic 
Ocean  and  Southern  Africa  ; 

1904,  September  9th,  noon,  in  South  America. 

This  series  is  remarkable  for  the  long  duration  of  total- 
ity, amounting  to  some  six  minutes. 

Let  us  now  consider  a  series  of  eclipses  recurring  at  reg- 
ular intervals  of  18  years  and  11  days.  Since  every  suc- 
cessive recurrence  of  such  an  eclipse  throws  the  conjunc- 
tion 28'  further  toward  the  west  of  the  node,  the  conjunc- 
tion must,  in  process  of  time,  take  place  so  far  back  from 
the  node  that  no  eclipse  will  occiir,  and  the  series  will  end. 
For  the  same  reason  there  must  be  a  commencement  to 
the  series,  the  first  eclipse  being  east  of  the  node.  A  new 
eclipse  thus  entering  will  at  first  be  a  very  small  one,  but 
will  be  larger  at  every  recurrence  in  each  Saros.  If  it  is 
an  eclipse  of  the  moon,  it  will  be  total  from  its  13th  until 
its  36th  recurrence.  There  will  then  be  about  13  partial 
eclipses,  each  of  which  will  be  smaller  than  the  last,  when 
they  will  fail  entirely,  the  conjunction  taking  place  so  far 
from  the  node  that  the  moon  does  not  touch  the  earth's 
shadow.  The  whole  interval  of  time  over  which  a  series 
of  lunar  eclipses  thus  extend  will  be  about  48  periods,  or 
865  years. 

When  a  series  of  solar  eclipses  begins,  the  penumbra  of 
the  first  will  just  graze  the  earth  not  far  from  one  of  the 
poles.  There  will  then  be,  on  the  average,  11  or  12  partial 
eclipses  of  the  sun,  each  larger  than  the  preceding  one, 
occurring  at  regular  intervals  of  one  Saros.  Then  the 
central  line,  whether  it  be  that  of  a  total  or  annular 
eclipse,  will  begin  to  touch  the  earth,  and  we  shall  have  a 
series  of  40  or  50  central  eclipses.  The  central  line  will 
strike  near  one  pole  in  the  first  part  of  the  series  ;  in  the 
equatorial  regions  about  the  middle  of  the  series,  and  will 


CHARACTERS  OF  ECLIPSES.  185 

leave  the  earth  by  the  other  pole  at  the  end.  Ten  or 
twelve  partial  eclipses  will  follow,  and  this  particular  se- 
ries will  cease.  The  whole  number  in  the  series  will  aver- 
age between  60  and  70,  occupying  a  few  centuries  over  a 
thousand  years. 

§  5.    CHARACTERS    OP   ECLIPSES. 

We  have  seen  that  the  possibility  of  a  total  eclipse  of  the  sun 
arises  from  the  occasional  very  slight  excess  of  the  apparent  angular 
diameter  of  the  moon  over  that  of  the  sun.  This  excess  is  so  slight 
that  such  an  eclipse  can  never  last  more  than  a  few  minutes.  It 
may  be  of  interest  to  point  out  the  circumstances  which  favor  a 
long  duration  of  totality.  These  are  : 

(1)  That  the  moon  should  be  as  near  as  possible  to  the  earth,  or, 
technically  speaking,  in   perigee,  because  its  angular  diameter  as 
seen  from  the  earth  will  then  be  greatest. 

(2)  That  the  sun  should  be  near  its  greatest  distance  from  the 
earth,  or  in  apogee,  because  then  its  angular  diameter  will  be  the 
least.     It  is  now  in  this  position  about  the  end  of  June  ;  hence  the 
most  favorable  time  for  a  total  eclipse  of  very  long  duration  is  in 
the  summer  months.     Since  the  moon  must  be  in  perigee  and  also 
between  the  earth  and  sun,  it  follows  that  the  longitude  of  the 
perigee  must  be  nearly  that  of  the  sun.     The  longitude  of  the  sun 
at  the  end  of  June  being  100°,  this  is  the  most  favorable  longi- 
tude of  the  moon's  perigee. 

(3)  The  moon  must  be  very  near  the  node  in  order  that  the  cen- 
tre of  the  shadow  may  fall  near  the  equator.  The  reason  of  this  con- 
dition is,  that  the  duration  of  a  total  eclipse  may  be  considerably 
increased  by  the  rotation  of  the  earth  on  its  axis.     We  have  seen 
that  the  shadow  sweeps  over  the  earth  from  west  toward  east  with  a 
velocity  of  about  3400  kilometres  per  hour.    Since  the  earth  rotates  in 
the  same  direction,  the  velocity  relative  to  the  observer  on  the  earth's 
surface  will  be  diminished  by  a  quantity  depending  on  this  velocity 
of  rotation,  and  therefore  greater,  the  greater  the  velocity.     The 
velocity  of   rotation  is  greatest  at   the  earth's  equator,   where  it 
amounts  to  1660  kilometres  per  hour,  or  nearly  half  the  velocity  of 
the  moon's  shadow.    Hence  the  duration  of  a  total  eclipse  may,  with- 
in the  tropics,  be  nearly  doubled  by  the  earth's  rotation.    When  all 
the  favorable  circumstances  combine  in  the  way  we  have  just  de- 
scribed, the  duration  of  a  total  eclipse  within  the  tropics  will  be 
about  seven  minutes  and  a  half.     In  our  latitude  the  maximum  du- 
ration will  be  somewhat  less,  or  not  far  from  six  minutes,  but  it  is 
only  on  very  rare  occasions,  hardly  once  in  many  centuries,  that  all 
these  favorable  conditions  can  be  expected  to  concur. 

Of  late  years,  solar  eclipses  have  derived  an  increased  in- 
terest from  the  fact  that  during  the  few  minutes  which 


186  ASTRONOMY. 

they  last  they  afford  unique  opportunities  for  investigating 
the  matter  which  lies  in  the  immediate  neighborhood  of 
the  sun.  Under  ordinary  circumstances,  this  matter  is 
rendered  entirely  invisible  by  the  effulgence  of  the  solar 
rays  which  illuminate  our  atmosphere  ;  but  when  a  body  so 
distant  as  the  moon  is  interposed  between  the  observer  and 
the  sun,  the  rays  of  the  latter  are  cut  off  from  a  region  a 
hundred  miles  or  more  in  extent.  Thus  an  amount  of 
darkness  in  the  air  is  secured  which  is  impossible  under 
any  other  circumstances  when  the  sun  is  far  above  the 
horizon.  Still  this  darkness  is  by  no  means  complete,  because 
the  sunlight  is  reflected  from  the  region  on  which  the  sun 
is  shining.  An  idea  of  the  amount  of  darkness  may  be 
gained  by  considering  that  the  face  of  a  watch  can  be  read 
during  an  eclipse  if  the  observer  is  careful  to  shade  his 
eyes  from  the  direct  sunlight  during  the  few  minutes  be- 
fore the  sun  is  entirely  covered  ;  that  stars  of  the  first 
magnitude  can  be  seen  if  one  knows  where  to  look  for 
them  ;  and  that  all  the  prominent  features  of  the  land- 
scape remain  plainly  visible.  An  account  of  the  investi- 
gations made  during  solar  eclipses  belongs  to  the  physical 
constitution  of  the  sun,  and  will  therefore  be  given  in  a 
subsequent  chapter. 

Occultation  of  Stars  by  the  Moon. — A  phenomenon 
which,  geometrically  considered,  is  analogous  to  an  eclipse 
of  the  sun  is  the  occultation  of  a  star  by  the  moon. 
Since  all  the  bodies  of  the  solar  system  are  nearer  than  the 
fixed  stars,  it  is  evident  that  they  must  from  time  to  time 
pass  between  us  and  the  stars.  The  planets  are,  however, 
so  small  that  such  a  passage  is  of  very  rare  occurrence, 
and  when  it  does  happen  the  star  is  generally  so  faint 
that  it  is  rendered  invisible  by  the  superior  light  of  the 
planet  before  the  latter  touches  it.  There  are  not  more 
than  one  or  two  instances  recorded  in  astronomy  of  a  well- 
authenticated  observation  of  an  actual  occultation  of  a  star 
by  the  opaque  body  of  a  planet,  although  there  are  several 
cases  in  which  a  planet  has  been  known  to  pass  over  a  star. 


OCCULTATION  OF  STARS.  187 

But  the  moon  is  so  large  and  her  angular  motion  so  rapid, 
that  she  passes  over  some  star  visible  to  the  naked  eye 
every  few  days.  Such  phenomena  are  termed  occultations 
of  stars  ty  the  moon.  It  must  not,  however,  be  supposed 
that  they  can  be  observed  by  the  naked  eye.  In  general, 
the  moon  is  so  bright  that  only  stars  of  the  first  magnitude 
can  be  seen  in  actual  contact  with  her  limb,  and  even  then 
the  contact  must  be  with  the  unillnminated  limb.  But 
with  the  aid  of  a  telescope,  and  the  predictions  given  in 
the  Ephemeris,  two  or  three  of  these  occultations  can  be 
observed  during  nearly  every  lunation. 


CHAPTER  VIII. 

THE   EAETH. 

OUR  object  in  the  present  chapter  is  to  trace  the  effects 
of  terrestrial  gravitation  and  to  study  the  changes  to 
which  it  is  subject  in  various  places.  Since  every  part 
of  the  earth  attracts  every  other  part  as  well  as  every 
object  upon  its  surface,  it  follows  that  the  earth  and 
all  the  objects  that  we  consider  terrestrial  form  a  sort 
of  system  by  themselves,  the  parts  of  which  are  firmly 
bound  together  by  their  mutual  attraction.  This  attrac- 
tion is  so  strong  that  it  is  found  impossible  to  project 
any  object  from  the  surface  of  the  earth  into  the  celestial 
spaces.  Every  particle  of  matter  now  belonging  to  the 
earth  must,  so  far  as  we  can  see,  remain  upon  it  forever. 

§   1.    MASS  AND  DENSITY  OP  THE    EARTH. 

We  begin  by  some  definitions  and  some  principles  re- 
specting attraction,  masses,  weight,  etc. 

The  mass  of  a  body  may  be  defined  as  the  quantity  of 
matter  which  it  contains. 

There  are  two  ways  to  measure  this  quantity  of  mat- 
ter :  (1)  By  the  attraction  or  weight  of  the  body — this 
weight  being,  in  fact,  the  mutual  force  of  attraction  be- 
tween the  body  and  the  earth  ;  (2)  By  the  inertia  of  the 
body,  or  the  amount  of  force  which  we  must  apply  to  it  in 
order  to  make  it  move  with  a  definite  velocity.  Mathe- 
matically, there  is  no  reason  why  these  two  methods  should 
give  the  same  result,  but  by  experiment  it  is  found  that 


MASS  OF  THE  EARTH.  189 

the  attraction  of  all  bodies  is  proportional  to  their  inertia. 
In  other  words,  all  bodies,  whatever  their  chemical  consti- 
tution, fall  exactly  the  same  number  of  feet  in  one  second 
under  the  influence  of  gravity,  supposing  them  in  a  vacu- 
um and  at  the  same  place  on  the  earth's  surface.  Although 
the  mass  of  a  body  is  most  conveniently  determined  by  its 
weight,  yet  mass  and  weight  must  not  be  confounded. 

The  weight  of  a  body  is  the  apparent  force  with  which 
it  is  attracted  toward  the  centre  of  the  earth.  As  we 
shall  see  hereafter,  this  force  is  not  the  same  in  all  parts  of 
the  earth,  nor  at  different  heights  above  the  earth's  sur- 
face. It  is  therefore  a  variable  quantity,  depending  upon 
the  position  of  the  body,  while  the  mass  of  the  body  is  re- 
garded as  something  inherent  in  it,  which  remains  constant 
wherever  the  body  may  be  taken,  even  if  it  is  carried 
through  the  celestial  spaces,  where  its  weight  would  be 
reduced  to  almost  nothing. 

The  unit  of  mass  which  we  may  adopt  is  arbitrary  ;  in 
fact,  in  different  cases  different  units  will  be  more  con- 
venient. Generally  the  most  convenient  unit  is  the  weight 
of  a  body  at  some  fixed  place  on  the  earth's  surface — the 
city  of  Washington,  for  example.  Suppose  we  take  such 
a  portion  of  the  earth  as  will  weigh  one  kilogram  in  Wash- 
ington, we  may  then  consider  the  mass  of  that  particular 
lot  of  earth  or  rock  as  a  kilogram,  no  matter  to  what  part 
of  the  universe  AVC  take  it.  Suppose  also  that  we  could 
bring  all  the  matter  composing  the  earth  to  the  city  of 
Washington,  one  kilogram  at  a  time,  for  the  purpose  of 
weighing  it,  returning  each  kilogram  to  its  place  in  the 
earth  immediately  after  weighing,  so  that  there  should  be 
no  disturbance  of  the  earth  itself.  The  sum  total  of  the 
weights  thus  found  would  be  the  mass  of  the  earth,  and 
would  be  a  perfectly  definite  quantity,  admitting  of  being 
expressed  in  kilograms  or  pounds.  We  can  readily  cal- 
culate the  mass  of  a  volume  of  water  equal  to  that  of  the 
earth  because  we  know  the  magnitude  of  the  earth  in 
litres,  and  the  mass  of  one  litre  of  water.  Dividing  this 


190  ASTRONOMY. 

into  the  mass  of  the  earth,  supposing  ourselves  able  to  de- 
termine this  mass,  and  we  shall  have  the  specific  gravity, 
or  what  is  more  properly  called  the  density  of  the  earth. 

What  we  have  supposed  for  the  earth  we  may  imagine 
for  any  heavenly  body — namely,  that  it  is  brought  to  the 
city  of  Washington  in  small  pieces,  arid  there  weighed  one 
piece  at  a  time.  Thus  the  total  mass  of  the  earth  or  any 
heavenly  body  is  a  perfectly  defined  and  determined 
quantity. 

It  may  be  remarked  in  this  connection  that  our  units  of 
weight,  the  pound,  the  kilogram,  etc.,  are  practically  units 
of  mass  rather  than  of  weight.  If  we  should  weigh  out 
a  pound  of  tea  in  the  latitude  of  Washington,  and  then 
take  it  to  the  equator,  it  would  really  be  less  heavy  at  the 
equator  than  in  Washington  ;  but  if  we  take  a  pound 
weight  with  us,  that  also  would  be  lighter  at  the  equator, 
so  that  the  two  would  still  balance  each  other,  and  the  tea 
would  be  still  considered  as  weighing  one  pound.  Since 
things  are  actually  weighed  in  this  way  by  weights  which 
weigh  one  unit  at  some  definite  place,  say  Washington, 
and  which  are  carried  all  over  the  world  without  being 
changed,  it  follows  that  a  body  which  has  any  given 
weight  in  one  place  will,  as  measured  in  this  way,  have 
the  same  apparent  weight  in  any  other  place,  although  its 
real  weight  will  vary.  But  if  a  spring  balance  or  any 
other  instrument  for  determining  actual  weights  were 
adopted,  then  we  should  find  that  the  weight  of  the  same 
body  varied  as  we  took  it  from  one  part  of  the  earth  to 
another.  Since,  however,  we  do  not  use  this  sort  of  an 
instrument  in  weighing,  but  pieces  of  metal  which  are 
carried  about  without  change,  it  follows  that  what  wre  call 
units  of  weight  are  properly  units  of  mass. 

Density  of  the  Earth. — We  see  that  all  bodies  around 
us  tend  to  fall  toward  the  centre  of  the  earth.  According 
to  the  law  of  gravitation,  this  tendency  is  not  simply  a 
single  force  directed  toward  the  centre  of  the  earth,  but 
is  the  resultant  of  an  infinity  of  separate  forces  arising  from 


MASS  OF  THE  EARTH.  191 

the  attractions  of  all  the  separate  parts  which  compose  the 
earth.  The  question  may  arise,  how  do  we  know  that  each 
particle  of  the  earth  attracts  a  stone  which  falls,  and  that 
the  whole  attraction  does  not  reside  in  the  centre  ?  The 
proofs  of  this  are  numerous,  and  consist  rather  in  the 
exactitude  with  which  the  theory  represents  a  great  mass 
of  disconnected  phenomena  than  in  any  one  principle  ad- 
mitting of  demonstration.  Perhaps,  however,  the  most 
conclusive  proof  is  found  in  the  observed  fact  that  masses 
of  matter  at  the  surface  of  the  earth  do  really  attract  each 
other  as  required  by  the  law  of  NEWTON.  It  is  found,  for 
example,  that  isolated  mountains  attract  a  plumb-line  in 
their  neighborhood.  The  celebrated  experiment  of  CAV- 
ENDISH wras  devised  for  the  purpose  of  measuring  the  at- 
traction of  globes  of  lead.  The  object  of  measuring  this 
attraction,  however,  was  not  to  prove  that  gravitation  re- 
sided in  the  smallest  masses  of  matter,  because  there  was 
no  doubt  of  that,  but  to  determine  the  mean  density  of  the 
earth,  from  which  its  total  mass  may  be  derived  by  simply 
multiplying  the  density  by  the  volume. 

It  is  noteworthy  that  though  astronomy  affords  us  the 
means  of  determining  with  great  precision  the  relative 
masses  of  the  earth,  the  moon,  and  all  the  planets,  it  does 
not  enable  us  to  determine  the  absolute  mass  of  any  hea- 
venly body  in  units  of  the  weights  we  use  on  the  earth. 
We  know,  for  instance,  from  astronomical  research,  that 
the  sun  has  about  328,000  times  the  mass  of  the  earth, 
and  the  moon  only  -£$  of  this  mass,  but  to  know  the  abso- 
lute mass  of  either  of  them  we  must  know  how  many 
kilograms  of  matter  the  earth  contains.  To  determine 
this,  we  must  know  the  mean  density  of  the  earth,  and  this 
is  something  about  which  direct  observation  can  give  us  no 
information,  because  we  cannot  penetrate  more  than  an 
insignificant  distance  into  the  earth's  interior.  The  only 
way  to  determine  the  density  of  the  earth  is  to  find  how 
much  matter  it  must  contain  in  order  to  attract  bodies  on 
Hs  surface  with  a  force  equal  to  their  observed  weight — 


192  ASTRONOMY. 

that  is,  with  such  intensity  that  at  the  equator  a  body  shall 
fall  nearly  ten  metres  in  one  second.  To  find  this  we 
must  know  the  relation  between  the  mass  of  a  body  and 
its  attractive  force.  This  relation  can  be  found  only  by 
measuring  the  attraction  of  a  body  of  known  mass.  An 
attempt  to  do  this  was  made  by  MASKELYNE,  Astronomer 
Royal  of  England,  toward  the  close  of  the  last  century, 
the  attracting  object  he  selected  being  Mount  Schehallien 
in  Scotland.  The  specific  gravity  of  the  rocks  composing 
this  mountain  was  well  enough  known  to  give  at  least  an 
approximate  result.  The  density  of  the  earth  thus  found 
was  4-71.  That  is,  the  earth  has  4.71  times  the  mass  of 
an  equal  volume  of  water.  This  result  is,  however,  un- 
certain, owing  to  the  necessary  uncertainty  respecting  the 
density  of  the  mountain  and  the  rocks  below  it. 

The  CAVENDISH  experiment  for  determining  the  attrac- 
tion of  a  pair  of  massive  balls  affords  a  much  more  perfect 
method  of  determining  this  important  element.  The 
most  careful  experiments  by  this  method  were  made  by 
BAILY  of  England  about  the  year  1845.  The  essential 
parts  of  the  apparatus  which  he  used  are  as  follows  : 

A  long  narrow  table  77bears  two  massive  spheVes  of  lead 
W  W,  one  at  each  end.  This  table  admits  of  being 
turned  around  on  a  pivot  in  a  horizontal  direction. 
Above  it  is  suspended  a  balance — that  is,  a  very  light  deal 
rod  e  with  a  weight  at  each  end  suspended  horizontally 
by  a  fine  silver  wire  or  fibre  of  silk  f  E.  The  weights  to 
be  attracted  are  attached  to  each  end  of  the  deal  rod.  The 
right-hand  one  is  visible,  while  the  other  is  hidden  be- 
hind the  left-hand  weight  W.  In  this  position  it  will  be 
seen  that  the  attraction  of  the  weights  W  tends  to  turn 
the  balance  in  a  direction  opposite  that  of  the  hands  of  a 
watch.  The  fact  is,  the  balance  begins  to  turn  in  this  di- 
rection, and  being  carried  by  its  own  momentum  beyond 
the  point  of  equilibrium,  comes  to  rest  by  a  twist  of  the 
thread.  It  is  then  carried  part  of  the  way  back  to  its 
original  position,  and  thus  makes  several  vibrations  which 


DENSITY  OF  THE  EARTH. 


193 


require  several  minutes.  At  length  it  comes  to  rest  in  a 
position  somewhat  different  from  its  original  one.  This 
position  and  the  times  of  vibration  are  all  carefully  noted. 
Then  the  table  T  is  turned  nearly  end  for  end,  so  that  one 
weight  W  shall  be  between  the  observer  and  the  right- 
hand  ball,  while  the  other  weight  is  beyond  the  left-hand 
ball,  and  the  observation  is  repeated.  A  series  of  observa- 
tions made  in  this  way  include  attractions  in  alternate  di- 


w 


FIG.  65. 

rections,  giving  a  result  from  which  accidental  errors  will 
be  very  nearly  eliminated. 

A  third  method  of  determining  the  density  of  the  earth 
is  founded  on  observations  of  the  change  in  the  intensity 
of  gravity  as  we  descend  below  the  surface  into  deep 
mines.  The  principles  on  which  this  method  rests  will  be 
explained  presently.  The  most  careful  application  of  it 
was  made  by  Professor  AIRY  in  the  Harton  Colliery,  Eng- 


194  ASTRONOMY. 

land.     The  results  of  this  and  the  other  methods  are  as 
follows  : 

CAVENDISH  and  HUTTON,  from  the  attraction  of  balls,  5  •  32 
KETCH,  "  "  "         5-58 

BAILY,  "  "  "        5-66 

MASKELYNE,  from  the  attraction  of  Schehallien 4-71 

AIRY,  from  gravity  in  the  Harton  Colliery 6-56 

Of  these  different  results,  that  of  BAILY  is  probably  the 
best,  and  the  most  probable  mean  density  of  the  earth  is 
about  5f  times  that  of  water.  This  is  more  than  double 
the  mean  speciiic  gravity  of  the  materials  which  compose 
the  surface  of  the  earth  ;  it  follows,  therefore,  that  the  in- 
ner portions  of  the  earth  are  much  more  dense  than  its 
outer  portions. 

§   2.    LAWS  OF  TERRESTRIAL  GRAVITATION. 

The  earth  being  very  nearly  spherical,  certain  theorems 
respecting  the  attraction  of  spheres  may  be  applied  to  it. 
The  fundamental  theorems  may  be  regarded  as  those 
which  give  the  attraction  of  a  spherical  shell  of  matter. 
The  demonstration  of  these  theorems  requires  the  use  of 
the  Integral  Calculus,  and  will  be  omitted  here,  only  the 
conditions  and  the  results  being  stated.  Let  us  then  im- 
agine a  hollow  shell  of  matter,  of  which  the  internal  and 
external  surfaces  are  both  spheres,  attracting  any  other 
masses  of  matter,  a  small  particle  we  may  suppose.  This 
particle  will  be  attracted  by  every  particle  of  the  shell 
with  a  force  inversely  as  the  square  of  its  distance  from  it. 
The  total  attraction  of  the  shell  will  be  the  resultant  of 
this  infinity  of  separate  attractive  forces.  Determining 
this  resultant  by  the  Integral  Calculus,  it  is  found  that  : 

Theorem  I.  — If  the  particle  he  outside  the  shell,  it  will 
le  attracted  as  if  the  whole  mass  of  the  shell  were  con- 
centrated in  its  centre. 

Theorem  II. — If  the  particle  he  inside  the  shell,  the  op~ 


ATTRACTION  OP  SPHERES.  105 

posite  attractions  in  every  direction  will  neutralize  each 
other,  no  matter  whereabouts  in  the  interior  the  particle 
nw,y  be,  and  the  resultant  attraction  of  the  shell  will  there- 
fore be  zero. 

To  apply  this  to  the  attraction  of  a  solid  sphere,  let  us 
first  suppose  a  body  either  outside  the  sphere  or  on  its  sur- 
face. If  we  conceive  the  sphere  as  made  up  of  a  great 
number  of  spherical  shells,  the  attracted  point  will  be  ex- 
ternal to  all  of  them.  Since  each  shell  attracts  as  if  its 
whole  mass  were  in  the  centre,  it 
follows  that  the  whole  sphere  at- 
tracts a  body  upon  the  outside  of 
its  surface  as  if  its  entire  mass 
were  concentrated  at  its  centre. 

Let  us  now  suppose  the  attract- 
ed particle  inside  the  sphere,  as 
at  /*,  Fig.  G6,  and  imagine  a 
spherical  surface  P  Q  concentric 
with  the  sphere  and  passing 
through  the  attracted  particle.  FlG*  66' 

All  that  portion  of  the  sphere  lying  outside  this  spherical 
surface  will  be  a  spherical  shell  having  the  particle  inside 
of  it,  and  will  therefore  exert  no  attraction  whatever  on 
the  particle.  That  portion  inside  the  surface  will  con- 
stitute a  sphere  with  the  particle  on  its  surface,  and  will 
therefore  attract  as  if  all  this  portion  were  concentrated 
in  the  centre.  To  find  what  this  attraction  will  be,  let  us 
first  suppose  the  whole  sphere  of  equal  density.  Let  us 
put 

&,  the  radius  of  the  entire  sphere. 

r,  the  distance  P  C  of  the  particle  from  the  centre. 
The  total  volume  of  matter  inside  the  sphere  P  Q  will 

then  be,  by  geometry,  -  n  r\     Dividing  by  the  square  of 

the  distance  /*,  we  see  that  the  attraction  will  be  repre- 
sented by 


106  ASTRONOMY. 

that  is,  inside  the  sphere  the  attraction  will  be  directly  as 
the  distance  of  the  particle  from  the  centre.  If  the  par- 
ticle is  at  the  surface  we  have  /•  —  a,  and  the  attraction  is 

4 

f*+ 

4. 
Outside  the  surface  the  whole  volume  of  the  sphere  -~  TT  a3 

o 

will  attract  the  particle,  and  the  attraction  will  be 

4       a3 

S*7 

If  we  put  r  =  a  in  this  formula,  we  shall  have  the  same 
result  as  before  for  the  surface  attraction. 

Let  us  next  suppose  that  the  density  of  the  sphere  va- 
ries from  its  centre  to  its  surface,  but  in  such  a  way  as  to 
be  equal  at  equal  distances  from  the  centre.  We  may 
then  conceive  of  it  as  formed  of  an  infinity  of  concentric 
spherical  shells,  each  homogeneous  in  density,  but  not  of 
the  same  density  with  the  others.  Theorems  I.  and  II. 
will  then  still  apply,  but  their  result  will  not  be  the  same 
as  in  the  case  of  a  homogeneous  sphere  for  a  particle  in- 
side the  sphere.  Referring  to  Fig.  66,  let  us  put 

Z>,  the  mean  density  of  the  shell  outside  the  particle  P. 
D',  the  mean  density  of  the  portion  P  Q  inside  of  P. 
We  shall  then  have  : 

Volume  of  the  shell,  -  TT  (a3  —  r3). 
o 

4 
Volume  of  the  inner  sphere,  --  TTT*. 

o 

Mass  of  the  shell  =  vol.  x  D  =  |  n  D  (a3  -  r3). 

o 

4. 
Mass  of   the  inner  sphere  =  vol.  x  D'  =  r-  TT  D'  r3. 

Mass  of  whole  sphere  =  sum  of  masses  of  shell  and  inner 
splere  =  -  n  (l)  a3  +  (D'  -  D)  />3). 


ATTRACTION  OF  SPHERES.  197 

Attraction  of  the  whole  sphere  upon  a  point  at  its  snr- 

Mass       4 


Attraction  of  the  inner  sphere  (the  same  as  that  of  the 

IVl  *m^         T~ 
whole  shell)  upon  a  point  at  P  =  —  —  =  «•  TT  D'  r. 

If,  as  in  the  case  of  the  earth,  the  density  continually  in- 
creases toward  the  centre,  the  value  of  D'  will  increase 
also  as  r  diminishes,  so  that  gravity  will  diminish  less 
rapidly  than  in  the  case  of  a  homogeneous  sphere,  and 
may,  in  fact,  actually  increase.  To  show  this,  let  us  sub- 
tract the  attraction  at  P  from  that  at  the  surface.  The 
difference  will  give  : 

Diminution  at  P  =  ^  TT  (D  a  +  (D'  -  D)  ~  -  D'  r)  . 
o       »  a  ' 

Now,  let  us  suppose  r  a  very  little  less  than  a,  and  put 

r  =  a  —  d  , 

d  will  then  be  the  depth  of  the  particle  below  the  surface. 
Cubing  this  value  of  r,  neglecting  the  higher  powers  of 
d,  and  dividing  by  «2,  we  find, 


Substituting  in  the  above  equation,  the  diminution  of  grav- 
ity at  P  becomes, 


We  see  that  if  3D  <  2Z>',  that  is,  if  the  density  at  the 
surface  is  less  than  f  of  the  mean  density  of  the  whole  in- 
ner mass,  this  quantity  will  become  negative,  showing  that 
the  force  of  gravity  will  be  less  at  the  surface  than  at  a 
small  depth  in  the  interior.  But  it  must  ultimately 
diminish,  because  it  is  necessarily  zero  at  the  centre. 
It  was  on  this  principle  that  Professor  Airy  determined 
the  density  of  the  earth  by  comparing  the  vibrations 


198  ASTRONOMY. 

of  a  pendulum  at  the  bottom  of  the  Harton  Colliery,  and 
at  the  surface  of  the  earth  in  the  neighborhood.  At  the 
bottom  of  the  mine  the  pendulum  gained  about  2s  •  5  per 
day,  showing  the  force  of  gravity  to  be  greater  than  at  the 
surface. 


§   3.    FIGURE  AND  MAGNITUDE    OP  THE  EARTH. 

If  the  earth  were  fluid  and  did  not  rotate  on  its  axis,  it 
would  assume  the  form  of  a  perfect  sphere.  The  opinion 
is  entertained  that  the  earth  was  once  in  a  molten  state, 
and  that  this  is  the  origin  of  its  present  nearly  spherical 
form.  If  we  give  such  a  sphere  a  rotation  upon  its  axis, 
the  centrifugal  force  at  the  equator  acts  in  a  direction  op- 
posed to  gravity,  and  thus  tends  to  enlarge  the  circle  of 
the  equator.  It  is  found  by  mathematical  analysis  that 
the  form  of  such  a  revolving  fluid  sphere,  supposing  it  to 
be  perfectly  homogeneous,  will  be  an  oblate  ellipsoid — that 
is,  all  the  meridians  will  be  equal  and  similar  ellipses,  hav- 
ing their  major  axes  in  the  equator  of  the  sphere  and  their 
minor  axes  coincident  with  the  axis  of  rotation.  Our  earth, 
however,  is  not  wholly  fluid,  and  the  solidity  of  its  conti- 
nents prevents  its  assuming  the  form  it  would  take  if  the 
ocean  covered  its  entire  surface.  When  we  speak  of  the  fig- 
ure of  the  earth,  we  mean,  not  the  outline  of  the  solid  and 
liquid  portions  respectively,  but  the  figure  which  it  would 
assume  if  its  entire  surface  were  an  ocean.  Let  us  imagine 
canals  dug  down  to  the  ocean  level  in  every  direction 
through  the  continents,  and  the  water  of  the  ocean  to  be 
admitted  into  them.  Then  the  curved  surface  touching 
the  water  in  all  these  canals,  and  coincident  with  the  sur- 
face of  the  ocean,  is  that  of  the  ideal  earth  considered  by 
astronomers.  By  the  figure  of  the  earth  is  meant  the 
figure  of  this  liquid  surface,  without  reference  to  the  in- 
equalities of  the  solid  surface. 

We  cannot  say  that  this  ideal  earth  is  a  perfect  ellipsoid, 
because  we  know  that  the  interior  is  not  homogeneous, 


MEASUREMENT  OF  THE  EARTH.  199 

but  all  the  geodetic  measures  heretofore  made  are  so  nearly 
represented  by  the  hypothesis  of  an  ellipsoid  that  the  lat- 
ter is  considered  as  a  very  close  approximation  to  the  true 
figure.  The  deviations  hitherto  noticed  are  of  so  irregu- 
|lar  a  character  that  they  have  not  yet  been  reduced  to  any 
certain  law.  The  largest  which  have  been  observed  seem 
to  be  due  to  the  attraction  of  mountains,  or  to  inequalities 
of  density  beneath  the  surface. 

Method  of  Triangulation. — Since  it  is  practically  im- 
possible to  measure  around  or  through  the  earth,  the  mag- 
nitude as  well  as  the  form  of  our  planet  has  to  be  found 
by  combining  measurements  on  its  surface  with  astronom- 
ical observations.  Even  a  measurement  on  the  earth's 
surface  made  in  the  usual  way  of  surveyors  would  be  im- 
practicable, owing  to  the  intervention  of  mountains,  rivers, 
forests,  and  other  natural  obstacles.  The  method  of  tri- 
angulatioii  is  therefore  universally  adopted  for  measure- 
ments extending  over  large  areas.  A  triangulation  is  ex- 
ecuted in  the  following  way  :  Two  points,  a  and  &,  a  few 


FlG.    67. — A  PART  OF   THE  FRENCH  TRTANGULATION  NEAR  PARIS. 

miles  apart,  are  selected  as  the  extremities  of  a  base-line. 
They  must  be  so  chosen  that  their  distance  apart  can  be 
accurately  measured  by  rods  ;  the  intervening  ground 
should  therefore  be  as  level  and  free  from  obstruction  as 
possible.  One  or  more  elevated  points,  EF,  etc.,  must 
be  visible  from  one  or  both  ends  of  the  base-line.  By 


200  ASTRONOMY. 

means  of  a  theodolite  and  by  observation  of  the  pole-star, 
the  directions  of  these  points  relative  to  the  meridian  are 
accurately  observed  from  each  end  of  the  base,  as  is  also 
the  direction  a  1)  of  the  base-line  itself.  Suppose  F  to 
be  a  point  visible  from  each  end  of  the  base,  then  in  the 
triangle  abJ^we  have  the  length  a  ~b  determined  by  actual 
measurement,  and  the  angles  at  a  and  Z>  determined  by  ob- 
servations. "With  these  data  the  lengths  of  the  sides  a  F 
and  I F  are  determined  by  a  simple  trigonometrical  com- 
putation. 

The  observer  then  transports  his  instruments  to  F,  and 
determines  in  succession  the  direction  of  the  elevated 
points  or  hills  D  E  G  H  J,  etc.  He  next  goes  in  succes- 
sion to  each  of  these  hills,  and  determines  the  direction  of 
all  the  others  which  are  visible  from  it.  Thus  a  network 
of  triangles  is  formed,  of  which  all  the  angles  are  observed 
with  the  theodolite,  while  the  sides  are  successively  calcu- 
lated trigonometrically  from  the  first  base.  For  instance, 
we  have  just  shown  how  the  side  a,F  is  calculated  ;  this 
forms  a  base  for  the  triangle  EFa,  the  two  remaining 
sides  of  which  are  computed.  The  side  EF  forms  the 
base  of  the  triangle  G  EF,  the  sides  of  which  are  calcu- 
lated, etc.  In  this  operation  more  angles  are  observed 
than  are  theoretically, necessary  to  calculate  the  triangles. 
This  surplus  of  data  serves  to  insure  the  detection  of  any 
errors  in  the  measures,  and  to  test  their  accuracy  by  the 
agreement  of  their  results.  Accumulating  errors  are  fur- 
ther guarded  against  by  measuring  additional  sides  from 
time  to  time  as  opportunity  offers. 

Chains  of  triangles  have  thus  been  measured  in  Russia 
from  the  Danube  to  the  Arctic  Ocean,  in  England  and 
France  from  the  Hebrides  to  Algiers,  in  this  country  down 
nearly  our  entire  Atlantic  coast  and  along  the  great  lakes, 
and  through  short  distances  in  many  other  countries. 
An  east  and  west  line  is  now  being  run  by  the  Coast  Sur- 
vey from  the  Atlantic  to  the  Pacific  Ocean.  Indeed  it 
may  be  expected  that  a  network  of  triangles  will  be  grad- 


MAGNITUDE  OF  THE  EARTH.  201 

ually  extended  over  the  surface  of  every  civilized  country, 
in  order  to  construct  perfect  maps  of  it. 

Suppose  that  we  take  two  stations  situated  north 
and  south  of  each  other,  determine  the  latitude  of  each, 
and  measure  the  distance  between  them.  It  is  evident  that 
by  dividing  the  distance  in  kilometres  by  the  difference  of 
latitude  in  degrees,  we  shall  have  the  length  of  one  degree 
of  latitude.  Then  if  the  earth  were  a  sphere,  we  should 
at  once  have  its  circumference  by  multiplying  the  length 
of  one  degree  by  360.  It  is  thus  found,  in  a  rough  way, 
that  the  length  of  a  degree  is  a  little  more  than  111  kilo- 
metres, or  between  69  and  70  English  statute  miles.  Its 
circumference  is  therefore  about  40,000  kilometres,  and 
its  diameter  between  12,000  and  13,000.* 

Owing  to  the  ellipticity  of  the  earth,  the  length  of  one 
degree  varies  with  the  latitude  and  the  direction  in  which 
it  is  measured.  The  next  step  in  the  order  of  accuracy  is 
to  find  the  magnitude  and  the  form  of  the  earth  from 
measures  of  long  arcs  of  latitude  (and  sometimes  of  longi- 
tude) made  in  different  regions,  especially  near  the  equa- 
tor and  in  high  latitudes.  But  we  shall  still  find  that  dif- 
ferent combinations  of  measures  give  slightly  different  re- 
sults, both  for  the  magnitude  and  the  ellipticity,  owing 
to  the  irregularities  in  the  direction  of  attraction  which  we 
have  already  described.  The  problem  is  therefore  to  find 
what  ellipsoid  will  satisfy  the  measures  with  the  least  sum 
total  of  error.  New  and  more  accurate  solutions  will  be 
reached  from  time  to  time  as  geodetic  measures  are  extend- 
ed over  a  wider  area.  The  following  are  among  the  most 
recent  results  hitherto  reached  :  LISTING  of  Gottingen 
in  1878  found  the  earth's  polar  semidiaineter,6355  •  270  kilo- 

*  When  the  metric  system  was  originally  designed  by  the  French,  it 
was  intended  that  the  kilometre  should  be  ro^o  of  the  distance  from 
the  pole  of  the  earth  to  the  equator.  This  would  make  a  degree  of  the 
meridian  equal,  on  the  average,  to  111$  kilometres.  But,  owing  to  the 
practical  difficulties  of  measuring  a  meridian  of  the  earth,  the  corre- 
spondence with  the  metre  actually  adopted  is  not  exact. 


202  ASTRONOMY. 

metres  ;  earth's  equatorial  semidiameter,  6377  •  377  kilo- 
metres ;  earth's  compression,  28\.6  of  the  equatorial  di- 
ameter ;  earth's  eccentricity  of  meridian,  0-08319.  An- 
other result  is  that  of  Captain  CLARKE  of  England,  who 
found  :  Polar  semidiameter,  6356  •  456  *  kilometres  ;  equa- 
torial semidiameter,  6378  •  191  kilometres. 

It  was  once  supposed  that  the  measures  were  slightly  bet- 
ter represented  by  supposing  the  earth  to  be  an  ellipsoid 
with  three  unequal  axes,  the  equator  itself  being  an  ellipse 
of  which  the  longest  diameter  was  500  metres,  or  about 
one  third  of  a  mile,  longer  than  the  shortest.  This  result 
was  probably  due  to  irregularities  of  gravity  in  those  parts 
of  the  continents  over  which  the  geodetic  measures  have 
extended  and  is  now  abandoned. 

Geographic  and  Geocentric  Latitudes. — An  obvious  re- 
sult of  the  ellipticity  of  the  earth  is  that  the  plumb-line 


FIG.  68. 

does  not  point  toward  the  earth's  centre.  Let  Fig.  68 
represent  a  meridional  section  of  the  earth,  JV  S  being  the 
axis  of  rotation,  E  Q  the  plane  of  the  equator,  and  0  the 
position  of  the  observer.  The  line  7/7?,  tangent  to  the 

*  Captain  Clarke's  results  are  given  in  feet,  the  polar  radius  being 
20,854,895  feet.  In  changing  to  metres,  the  logarithm  of  the  factor  has 
been  taken  as  9.4840071. 


FORCE  OF  GRAVITY.  203 

earth  at  0,  will  then  represent  the  horizon  of  the  observer, 
while  the  line  ZJV',  perpendicular  to  II R,  and  therefore 
normal  to  the  earth  at  Q,  will  be  vertical  as  determined 
by  the  plumb-line.  The  angle  ON'  Q,  or  ZO  Q',  which 
the  observer's  zenith  makes  with  the  equator,  will  then  be 
his  astronomical  or  geographical  latitude.  This  is  the  lat- 
itude which  in  practice  we  nearly  always  have  to  use,  be- 
cause we  are  obliged  to  determine  latitude  by  astronomical 
observation,  and  not  by  measurement  from  the  equator. 
AV  e  cannot  determine  the  direction  of  the  true  centre  C  of 
the  earth  by  direct  observation  of  any  kind,  but  only  that 
of  the  plumb-line,  or  of  the  perpendicular  to  a  fluid  sur- 
face. Z  0  Q'  is  therefore  the  astronomical  latitude.  If, 
however,  we  conceive  the  line  0 ' Oz  drawn  from  the  cen- 
tre of  the  earth  through  6>,  z  will  be  the  observer's  geo- 
centric zenith,  while  the  angle  0  C  Q  will  be  his  geocen- 
tric latitude.  It  will  be  observed  that  it  is  the  geocentric 
and  not  the  geographic  latitude  which  gives  the  true  posi- 
tion of  the  observer  relative  to  the  earth's  centre.  The 
difference  between  the  two  latitudes  is  the  angle  CON1 
or  Z  0  z  ;  this  is  called  the  angle  of  the  vertical.  It  is  zero 
at  the  poles  and  at  the  equator,  because  here  the  normals 
pass  through  the  centre  of  the  ellipse,  and  it  attains  its 
maximum  of  11'  30"  at  latitude  45°.  It  will  be  seen  that 
the  geocentric  latitude  is  always  less  than  the  geographic. 
In  north  latitudes  the  geocentric  zenith  is  south  of  the  ap- 
parent zenith  and  in  southern  latitudes  north  of  it,  being 
nearer  the  equator  in  each  case. 


§  4.     CHANGE    OP    GRAVITY    WITH    THE    LATI- 
TUDE. 

If  the  earth  were  a  perfect  sphere,  and  did  not  rotate  on  its  axis,  the 
intensity  of  gravity  would  be  the  same  over  its  entire  surface.  There 
is  a  slight  variation  from  two  causes,  namely,  (1)  The  elliptic  form 
of  our  globe,  and  (2)  the  centrifugal  force  generated  by  its  rotation 
on  its  axis.  Strictly  speaking,  the  latter  is  not  a  change  in  the 
real  force  of  gravity,  or  of  the  earth's  attraction,  but  only  an 
apparent  force  of  another  kind  acting  in  opposition  to  gravity. 


204  ASTRONOMY. 

The  intensity  of  gravity  is  measured  by  the  distance  which  a 
heavy  body  in  a  vacuum  will  fall  in  a  unit  of  time,  say  one  second. 
Either  10  metres  or  32  feet  may  be  regarded  as  a  rough  approxima- 
tion to  its  value.  There  are,  however,  so  many  practical  difficul- 
ties in  the  way  of  measuring  with  precision  the  distance  a  body 
falls  in  one  second,  that  the  force  of  gravity  is,  in  practice,  deter- 
mined indirectly  by  finding  the  length  of  the  second's  pendulum. 
It  is  shown  in  mechanics  that  if  a  pendulum  of  length  L  vibrates 
in  a  time  T,  a  heavy  body  will  in  this  time  T  fall  through  the 
space  7T2  Z,  TT  being  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter.  (7r=3- 14159  .  .  .  7r2= 9 -869604.)  Therefore,  to  find  the 
force  of  gravity  we  have  only  to  determine  the  length  of  the 
second's  pendulum,  and  multiply  it  by  this  factor. 

The  determination  of  the  mean  attractive  force  of  the  earth  is 
important  in  order  that  we  may  compute  its  action  on  the  moon 
and  other  heavenly  bodies,  while  the  variations  of  this  attraction 
afford  us  data  for  judging  of  the  variations  of  density  in  the  earth's 
interior.  Scientific  expeditions  have  therefore  taken  pains  to 
determine  the  length  of  the  second's  pendulum  at  numerous  points 
on  the  globe.  To  do  this,  it  is  not  necessary  that  they  should 
actually  measure  the  length  of  the  pendulum  at  all  the  places  they 
visit.  They  have  only  to  carry  some  one  pendulum  of  a  very  solid 
construction  to  each  point  of  observation,  and  observe  how  many 
vibrations  it  makes  in  a  day.  They  know  that  the  force  of  gravity 
is  proportional  to  the  square  of  the  number  of  vibrations.  Before 
and  after  the  voyage,  they  count  the  vibrations  at  some  standard 
point — London  for  instance.  Thus,  by  simply  squaring  the  number 
of  vibrations  and  comparing  the  squares,  they  have  the  ratio 
which  gravity  at  various  points  of  the  earth's  surface  bears  to 
gravity  at  London.  Tt  is  then  only  necessary  to  determine  the 
absolute  intensity  of  gravity  at  London  to  infer  it  at  all  the 
other  points  for  which  the  ratio  is  known.  From  a  great  number 
of  observations  of  this  kind,  it  is  found  that  the  length  of  the 
second's  pendulum  in  latitude  0  may  be  nearly  represented  by  the 
equation, 

L  =  Om  •  99099  (1  +  0  •  00520  sin2  0). 

From  this,    the    force  of    gravity  is   found  by  multiplying    by 
7T2  =  9  •  8696,  giving  the  result  : 

g'  =  9m  •  7807  (1  +  0  •  00520  sin2  0). 

These  formulae  show  that  the  apparent  force  of  gravity  increases 
by  a  little  more  than  ^  of  its  whole  amount  from  the  equator  to 
the  poles.  We  can  readily  calculate  how  much  of  the  diminution 
at  the  equator  is  due  to  the  centrifugal  force  of  the  earth's  rotation. 
By  the  formulae  of  mechanics,  the  centrifugal  force  is  given  by  the 
equation, 

4^r2r 

J       '      rri-i    ) 


TERRESTRIAL  GRAVITY.  205 

T  being  the  time  of  one  revolution,  and  r  the  radius  of  the  circle  of 
rotation.  Supposing  the  earth  a  sphere,  which  will  cause  no 
important  error  in  our  present  calculation,  the  distance  of  a  point 
on  the  earth's  surface  in  latitude  <t>  from  the  axis  of  rotation  of  the 
earth  is, 

r  =  a  cos  0, 

a  being  the  earth's  radius.  The  centrifugal  force  in  latitude  0  is 
therefore 

_  4  7T2  a  cos  (j> 

J   ~  yra  • 

But  this  force  does  not  act  in  the  direction  normal  to  the  earth's 
surface,  but  perpendicular  to  the  axis  of  the  earth,  which  direction 
makes  the  angle  </>  with  the  normal.  We  may  therefore  resolve  the 
force  into  two  components,  one,  /  sin  0,  along  the  earth's  surface 
toward  the  equator,  the  other,/  cos  0,  downward  toward  its  centre. 
The  first  component  makes  the  earth  a  prolate  ellipsoid,  as  already 
shown,  while  the  second  acts  in  opposition  to  gravity.  The  cen- 
trifugal force,  therefore,  diminishes  gravity  by  the  amount, 

4  7T2  a  cos2 


7\  the  sidereal  day,  is  86,164  seconds  of  mean  time,  while  a,  for 
the  equator,  is  6,377,377  metres.  Substituting  in  this  expression, 
the  centrifugal  force  becomes 

/cos  tf  =  Om  •  03391  cos2  0  =  Om  •  03391  (1  —  sin2  0), 

or  at  the  equator  a  little  more  than  yfa  the  force  of  gravity.  The 
expression  for  the  apparent  force  of  gravity  given  by  observation, 
which  we  have  already  found,  may  be  put  in  the  form, 

g'  =9m-7807  +  Om-  05087  sin8  0. 

This  is  the  true  force  of  gravity  diminished  by  the  centrifugal 
force  ;  therefore,  to  find  that  true  force  we  must  add  the  centri- 
fugal force  to  it,  giving  the  result  : 

g  —  9m  •  8146  +  Om  •  01696  sin2  0 
=  9m-8146  (1  +  0-001728  sin2  0), 

for  the  real  attraction  of  the  spheroidal  earth  upon  a  body  on  its 
surface  in  latitude  </>. 

It  will  be  interesting  to  compare  this  result  with  the  attraction 
of  a  spheroid  having  the  same  ellipticity  as  the  earth.  It  is  found 
by  integration  that  if  0,  supposed  small,  be  the  eccentricity  of  a 
homogeneous  oblate  ellipsoid,  and  g0  its  attraction  upon  a  body 
on  its  equator,  its  attraction  at  latitude  £  will  be  given  by  the 
equation, 


206  ASTRONOMY. 

In  the  case  of  the  earth,  e  =  0-0817  ;    ^e"2  =  0-000667  ;    so  that 
the  expression  for  gravity  would  be, 

g  =  00  (1  +  0-000667  sin». 

We  see  that  the  factor  of  sin2  d,  which  expresses  the  ratio  in 
which  gravity  at  the  poles  exceeds  that  at  the  equator,  has  less  than 
half  the  value  (-001780),  which  we  have  found  from  observation. 
This  difference  arises  from  the  fact  that  the  earth  is  not  homogene- 
ous, but  increases  in  density  from  the  surface  toward  the  centre. 
To  see  how  this  result  follows,  let  us  first  inquire  how  the  eartjh 
would  attract  bodies  where  its  surface  now  is  if  its  whole  mass 
were  concentrated  in  its  centre.  The  distance  of  the  equat/r 
frorn^he  centre  is  to  that  of  the  poles  from  the  centre  as  1  jb 
Vl  —e*.  Therefore,  in  the  case  supposed,  attraction  at  the  equa*  r 
would  be  to  attraction  at  the  poles  as  1— e2  to  1.  The  ratio  of  - 
crease  of  attraction  at  the  poles  is  therefore  in  this  extreme  cfv  e 
about  ten  times  what  it  is  for  the  homogeneous  ellipsoid.  We  c'^- 
clude,  therefore,  that  the  more  nearly  the  earth  approaches  t\is 
extreme  case — that  is,  the  more  it  increases  in  density  toward  the 
centre — the  greater  will  be  the  difference  of  attraction  at  the  po*les 
and  the  equator. 


§    5.     MOTION    OF    THE    EARTH'S    AXIS,    OB    PRE- 
CESSION OF  THE  EQUINOXES. 

Sidereal  and  Equinoctial  Year. — In  describing  the  <ip- 
parent  motion  of  the  sun,  two  ways  were  shown  of  find- 
ing the  time  of  its  apparent  revolution  ^around  the  sphere 
—in  other  words,  of  fixing  the  length  of  a  year.  One  of 
these  methods  consists  in  finding  the  interval  between  suc- 
cessive passages  through  the  equinoxes,  or,  which  is  the 
same  thing,  across  the  plane  of  the  equator,  and  the  other 
by  finding  when  it  returns  to  the  same  position  among 
the  stars.  Two  thousand  years  ago,  HIPPARCHUS  found, 
by  comparing  his  own  observations  with  those  made  two 
centuries  before  by  TIMOCHAEIS,  that  these  two  methods 
of  fixing  the  length  of  the  year  did  not  give  the  same 
result.  It  had  previously  been  considered  that  the  length 
of  a  year  was  about  365  J  days,  and  in  attempting  to  correct 
this  period  by  comparing  his  observed  times  of  the  sun's 
passing  the  equinox  with  those  of  TIMOCHARIS,  HIPPAR- 
CHUS found  that  it  required  a  diminution  of  seven  or  eight 


LENGTH  OF  THE   YEAR.  207 

/ 

minutes.  He  therefore  concluded  that  the  true  length  of 
the  equinoctial  year  was  365  days,  5  hours,  and  about  53 
minutes.  When,  however,  he  considered  the  return,  not 
to  the  equinox,  but  to  the  same  position  relative  to  the 
bright  star  Spica  Virginis,  he  found  that  it  took  some 
minutes  more  than  365J  days  to  complete  the  revolution. 
Thus  there  are  two  years  to  be  distinguished,  the  tropical 
jr  equinoctial  year  and  the  sidereal  year.  The  first  is 

neasured  by  the  time  of  the  earth's  return  to  the  equinox  ; 

he  second  by  its  return  to  the  same  position  relative  to  the 
*4ars.  Although  the  sidereal  year  is  the  correct  astronom- 
ical period  of  one  revolution  of  the  earth  around  the  sun, 
/£t  the  equinoctial  year  is  the  one  to  be  used  in  civil  life, 
because  it  is  upon  that  year  that  the  change  of  seasons 
depends.  Modern  determinations  show  the  respective 
lengths  of  the  two  years  to  be  : 

Sidereal  year,         365d  6h    9m    9s  =  365d  •  25636. 
Equinoctial  year,  365d  5h  48m  46s  ==  365d-  24220. 

It  is  evident  from  this  difference  between  the  two  years 
that  the  position  of  the  equinox  among  the  stars  must  be 
changing,  and  must  move  toward  the  west,  because  the 
equinoctial  year  \:  the  shorter.  This  motion  is  called  the 
precession  of  the  equinoxes ,  and  amounts  to  about  50* 
per  year.  The  equinox  being  simply  the  point  in  which 
the  equator  and  the  ecliptic  intersect,  it  is  evident  that  it 
can  change  only  through  a  change  in  one  or  both  of  these 
circles.  HIPPARCHUS  found  that  the  change  was  in  the 
equator,  and  not  in  the  ecliptic,  because  the  declinations  of 
the  stars  changed,  while  their  latitudes  did  not.*  Since 

*  To  describe  the  theory  of  the  ancient  astronomers  with  perfect 
correctness,  we  ought  to  say  that  they  considered  the  planes  both  of  the 
equator  and  ecliptic  to  be  invariable  and  the  motion  of  precession  to 
be  due  to  a  slow  revolution  of  the  whole  celestial  sphere  around  the 
pole  of  the  ecliptic  as  an  axis.  This  would  produce  a  change  in  the 
position  of  the  stars  relative  to  the  equator,  but  not  relative  to  the 
ecliptic. 


208  ASTRONOMY. 

the  equator  is  defined  as  a  circle  everywhere  90°  distant 
from  the  pole,  and  since  it  is  moving  among  the  stars,  it 
follows  that  the  pole  must  also  be  moving  among  the  stars. 
But  the  pole  is  nothing  more  than  the  point  in  which  the 
earth's  axis  of  rotation  intersects  the  celestial  sphere  :  it 
must  be  remembered  too  that  the  position  of  this  pole7^ 
the  celestial  sphere  depends  solely  upon  the  direction  oi 
the  earth's  axis,  and  is  not  changed  by  the  motion  of  the 
earth  around  the  sun,  because  the  sphere  is  considered  to 
be  of  infinite  radius.  Hence  precession  shows  that  the 
direction  of  the  earth's  axis  is  continually  changing. 
Careful  observations  from  the  time  of  HIPPAKCHUS  until 
now  show  that  the  change  in  question  consists  in  a  slow 
revolution  of  the  pole  of  the  earth  around  the  pole  of  the 
ecliptic  as  projected  on  the  celestial  sphere.  The  rate  of 
motion  is  such  that  the  revolution  will  be  completed  in 
between  25,000  and  26,000  years.  At  the  end  of  this 
period  the  equinox  and  solstices  will  have  made  a  com- 
plete revolution  in  the  heavens. 

The  nature  of  this  motion  will  be  seen  more  clearly  by  referring 
to  Fig.  46,  p.  109.  We  have  there  represented  the  earth  in  four 
positions  during  its  annual  revolution.  We  have  represented  the  axis 
as  inclining  to  the  right  in  each  of  these  positions,  and  have  de- 
scribed it  as  remaining  parallel  to  itself  during  an  entire  revolution. 
The  phenomena  of  precession  show  that  this  is  not  absolutely  true, 
but  that,  in  reality,  the  direction  of  the  axis  is  slowly  changing. 
This  change  is  such  that,  after  the  lapse  of  some  6400  years,  the 
north  pole  of  the  earth,  as  represented  in  the  figure,  will  not  in- 
cline to  the  right,  but  toward  the  observer,  the  amount  of  the  in- 
clination remaining  nearly  the  same.  The  result  will  evidently  be 
a  shifting  of  the  seasons.  At  D  we  shall  have  the  winter  solstice, 
because  the  north  pole  will  be  inclined  toward  the  observer  and 
therefore  from  the  sun,  while  at  A  we  shall  have  the  vernal  equinox 
instead  of  the  winter  solstice,  and  so  on. 

In  6400  years  more  the  north  pole  will  be  inclined  toward  the 
left,  and  the  seasons  will  be  reversed.  Another  interval  of  the 
same  length,  and  the  north  pole  will  be  inclined  from  the  observer, 
the  seasons  being  shifted  through  another  quadrant.  Finally,  at 
the  end  of  about  25, 800  years,  the  axis  will  have  resumed  its  original 
direction. 

Precession  thus  arises  from  a  motion  of  the  earth  alone,  and 
not  of  the  heavenly  bodies.  Although  the  direction  of  the  earth's 
axis  changes,  yet  the  position  of  this  axis  relative  to  the  crust  of  the 


PRECESSION.  209 

earth  remains  invariable.  Some  have  supposed  that  precession 
would  result  in  a  change  in  the  position  of  the  north  pole  on  the 
surface  of  the  earth,  so  that  the  northern  regions  would  be  covered 
by  the  ocean  as  a  result  of  the  different  direction  in  which  the 
ocean  would  be  carried  by  the  centrifugal  force  of  the  earth's  rota- 
tion. This,  however,  is  a  mistake.  It  has  been  shown  by  a  mathe- 
matical investigation  that  the  position  of  the  poles,  and  therefore 
ijf  the  equator,  on  the  surface  of  the  earth,  cannot  change  except 
.  from  some  variation  in  the  arrangement  of  the  earth's  interior. 
Scientific  investigation  has  yet  shown  nothing  to  indicate  any  prob- 
ability of  such  a  change. 

The  motion  of  precession  is  not  uniform,  but  is  subject  to  several 
inequalities  which  are  called  Nutation.  These  can  best  be  under- 
stood in  connection  with  the  forces  which  produce  precession. 

Cause  of  Precession,  etc. — Sir  ISAAC  NEWTON  showed  that  pre- 
cession was  due  to  an  inequality  in  the  attraction  of  the  sun  and 
moon  produced  by  the  spheroidal  figure  of  the  earth.  If  the  earth 
were  a  perfect  homogeneous  sphere,  the  direction  of  its  axis  would 


FIG.  69. 

never  change  in  consequence  of  the  attraction  of  another  body. 
But  the  excess  of  matter  around  the  equatorial  regions  of  the  earth 
is  attracted  by  the  sun  and  moon  in  such  a  way  as  to  cause  a  turn- 
ing force  which  tends  to  change  the  direction  of  the  axis  of  rota- 
tion. To  show  the  mode  of  action  of  this  force,  let  us  consider  the 
earth  as  a  sphere  encircled  by  a  large  ring  of  matter  extending 
around  its  equator,  as  in  Fig.  69.  Suppose  a  distant  attracting  body 
situated  in  the  direction  (7c,  so  that  the  lines  in  which  the  parts  of 
the  ring  are  attracted  are  A  a,  B1>,  Cc,  etc.,  which  will  be  nearly 
parallel.  The  attractive  force  will  gradually  diminish  from  A  to 
B,  owing  to  the  greater  distance  of  the  latter  from  the  attracting 
body.  Let  us  put  : 

r,  the  distance  of  the  centre  C  from  the  attracting  body, 
p,  the  radius  A  G  =  B  G  of  the  equatorial  ring,  multiplied  by  the 
cosine  of  the  angle  A  C  c,  so  that  the  distance  of  A  from  the  attract- 
ing centre  is  r— p,  and  that  of  #is  r  +  p. 
m,  the  mass  of  the  attracting  body  ; 


210  ASTRONOMY. 

The  accelerative  attraction  exerted  at  the  three  points  A,C,B  will 
then  be 


(r-Py      r"    (r+p)2' 

The  radius  p  being  very  small  compared  with  r,  we  may  develop  the 
denominators  of  the  first  and  third  fractions  in  powers  of  - 

by  the  binomial  theorem,  and  neglect  all  powers  after  the  first. 
The  attractions  will  then  be  approximately  : 

m       2m  p      m      m      2mp 

P  +  -pr  >-'js\'  ?'    -jr- 

The  forces  — -—  will  be  very  small  compared  with  ™  on  account 
of  the  smallness  of  p. 
The  principal  force  —  will  cause  all  parts  of  the  body  to  fall 

7 

equally  toward  the  attracting  centre,  and  will  therefore  cause  no 
rotation  in  the  body  and  no  change  in  the  direction  of  the  axis  N 8. 
Supposing  the  body  to  revolve  around  the  centre  in  an  orbit,  we 
may  conceive  this  attraction  to  be  counterbalanced  by  the  so-called 
centrifugal  force.* 

Subtracting  this  uniform  principal  force,  there  is  left  a  force  - 

acting  on  A  in  the  direction  A  a,  and  an  equal  force  acting  on  B  in 
the  opposite  direction  ft  B.  It  is  evident  that  these  two  forces  tend 
to  make  the  earth  rotate  around  an  axis  passing  through  C  in  such 
a  direction  as  to  make  the  line  C  A  m  coincide  with  C  c  ,  and  that, 
if  no  cause  modified  the  action  of  these  forces,  the  earth  would  os- 
cillate back  and  forth  on  that  axis. 

*  We  may  here  mention  a  very  common  misapprehension  respecting 
what  is  sometimes  called  centrifugal  force,  and  is  supposed  to  be  a 
force  tending  to  make  a  body  fly  away  from  the  centre.  It  is  some- 
times said  that  the  body  will  fly  from  the  centre  when  the  centrifugal 
force  exceeds  the  centripetal,  and  toward  it  in  the  opposite  case.  This  is 
a  mistake,  such  a  force  as  this  having  no  existence.  The  so-called 
centrifugal  force  is  not  properly  a  centrifugal  force  at  all.  but  only  the 
reaction  of  the  whirling  body  against  the  centripetal  force,  which,  by  the 
third  law  of  motion,  is  equal  and  opposite  to  that  force.  When  a  stone 
is  whirled  in  a  sling  the  tension  on  the  string  is  simply  the  force  neces- 
sary to  make  the  stone  constantly  deviate  'from  the  straight  line  in 
which  it  tends  to  move,  and  is  the  same  as  the  resistance  which  the 
stone  offers  to  this  deviation  in  consequence  of  its  inertia.  So,  in  the 
case  of  the  planets,  the  centrifugal  force  is  only  the  resistance  offered 
by  the  inertia  of  the  planet  to  the  sun's  attraction.  If  the  sling  should 
break,  or  if  the  sun  should  cease  to  attract  the  planet,  the  centripetal 
and  centrifugal  forces  would  both  cease  instantly,  and  the  stone  or 
planet  would,  in  accordance  with  the  first  law  of  motion,  fly  forward 
in  the  straight  line  in  which  it  was  moving  at  the  moment. 


NUTATION.  211 

But  a  modifying  cause  is  found  in  the  rotation  of  the  earth  on  its 
own  axis,  which  prevents  any  change  in  the  angle  m  C  c  ,  but 
causes  a  very  slow  revolution  of  the  axis  N  8  around  the  perpen- 
dicular line  C  E,  which  motion  is  that  of  precession.* 

Nutation. — It  will  he  seen  that,  under  the  influence  of  the  grav- 
itation of  the  sun  and  moon,  precession  cannot  be  uniform.  At  the 
time  of  the  equinoxes  the  equator  A  B  of  the  earth  passes  through 
the  sun,  and  the  latter  lies  in  the  line  B  C  A  m,  so  that  the  small 
processional  force  tending  to  displace  the  equator  must  then  vanish. 
This  force  increases  on  both  sides  of  the  equinox,  and  attains  a 
maximum  at  the  solstices  when  the  angle  m  C  c  is  23|°.  Hence  the 
precession  produced  by  the  sun  takes  place  by  semi-annual  steps. 
One  of  these  steps,  however,  is  a  little  longer  than  the  other, 
because  the  earth  is  nearer  the  sun  in  December  than  in  June. 

Again,  we  have  seen  that  the  inclination  of  the  moon's  orbit  to 
the  equator  ranges  from  18£'  to  28£°  in  a  period  of  18-6  years. 
Since  the  precessional  force  depends  on  this  inclination,  the 
amount  of  precession  due  to  the  action  of  the  moon  has  a  period 
equal  to  one  revolution  of  the  moon's  node,  or  18*6  years.  These 
inequalities  in  the  motion  of  precession  are  termed  nutation. 

Changes  in  the  Right  Ascensions  and  Declinations  of 
the  Stars. — Since  the  declination  of  a  heavenly  body  is  its  an- 
gular distance  from  the  celestial  equator,  it  is  evident  that  any 
change  in  the  position  of  the  equator  must  change  the  declinations 
of  the  fixed  stars.  Moreover,  since  right  ascensions  are  counted 
from  the  position  of  the  vernal  equinox,  the  change  in  the  position 
of  this  equinox  produced  by  precession  and  nutation  must  change 
the  right  ascensions  of  the  stars.  The  motion  of  the  equator  may 
be  represented  by  supposing  it  to  turn  slowly  around  an  axis  lying 
in  its  plane,  and  pointing  to  6h  and  18h  of  right  ascension.  All 
that  section  of  the  equator  lying  within  6h  of  the  vernal  equinox 
(see  Fig.  45,  page  103)  is  moving  toward  the  south  (downward  in 
the  figure),  while  the  opposite  section,  from  6h  to  18h  right  ascen- 
sion, is  moving  north.  The  amount  of  this  motion  is  20"  annually. 
It  is  evident  that  this  motion  will  cause  both  equinoxes  to  shift 
toward  the  right,  and  the  geometrical  student  will  be  able  to  see 
that  the  amount  of  the  shift  will  be  : 

*  The  reason  of  this  seeming  paradox  is  that  the  rotative  forces  acting 
on  A  and  B  are  as  it  were  distributed  by  the  diurnal  rotation  around 
W8.  Suppose,  for  example,  that  A  receives  a  down  ward  and  B  an  up- 
ward impulse,  so  that  they  begin  to  move  in  these  directions.  At  the 
end  of  twelve  hours  A  has  moved  around  to  B,  so  that  its  downward 
motion  now  tends  to  increase  the  angle  m  C  c,  and  the  upward  motion  of 
B  has  the  same  effect.  If  we  suppose  a  series  of  impulses,  a  diminution 
of  the  inclination  will  be  produced  during  the  first  12  hours,  but  after 
that  the  effect  of  each  impulse  will  be  counterbalanced  by  that  of  12 
hours  before,  so  that  no  further  diminution  will  take  place  ;  but 
every  impulse  will  produce  a  sudden  permanent  change  in  the  direction 
of  the  axis  JV  8,  the  end  N  moving  toward  and  8  from  the  observer. 

This  same  law  of  rotation  is  exemplified  in  the  gyroscope  and  the 
child's  top,  each  of  which  are  kept  erect  by  the  rotation,  though  grav- 
ity tends  to  make  them  fall. 


312  ASTRONOMY 

On  the  equator,  20"  cot  «  ; 

On  the  ecliptic,  20*  cosec  w  ; 

w  being  the  obliquity  of  the  ecliptic  (23°  27|').  In  consequence, 
the  right  Ascensions  of  stars  near  the  equator  are  constantly  increas- 
ing by  about  46"  or  arc,  or  3s. 07  of  time  annually.  Away  from 
the  equator  the  increase  will  vary  in  amount,  because,  owing  to  the 
motion  of  the  pole  of  the  earth,  the  point  in  which  the  equator  is 
intersected  by  the  great  circle  passing  through  the  pole  and  the 
star  will  vary  as  well  as  the  equinox,  it  being  remembered  that  the 
right  ascension  of  the  star  is  the  distance  of  this  point  of  intersec- 
tion from  the  equinox. 

The  adept  in  spherical  trigonometry  will  find  it  an  improving 
exercise  to  work  out  the  formula  for  the  annual  change  in  the  right 
ascension  and  declination  of  the  stars,  arising  from  the  motion  of 
the  equator,  and  consequently  of  the  equinox.  He  will  find  the 
result  to  be  as  follows  :  Put 

?i,  the  annual  angular  motion  of  the  equator  (20" -06), 

w,  its  obliquity  (23°  27' -5), 

a  A,  the  right  ascension  and  declination  of  the  star ; 

Then  we  shall  find  : 

Annual  change  in  R.  A.  =  n  cot  u  +  n  sin  <x.  tan  6, 

Annual  change  in  Dec.  =  n  cos  a. 


CHAPTER  IX. 

CELESTIAL   MEASUREMENTS    OF   MASS   AND 
DISTANCE. 

§    1.    THE  CELESTIAL  SCALE    OF  MEASUREMENT. 

THE  units  of  length  and  mass  employed  by  astronomers 
are  necessarily  different  from  those  used  in  daily  life. 
For  instance,  the  distances  and  magnitudes  of  the  heavenly 
bodies  are  never  reckoned  in  miles  or  other  terrestrial 
measures  for  astronomical  purposes  ;  when  so  expressed 
it  is  only  for  the  purpose  of  making  the  subject  clearer  to 
the  general  reader.  The  units  of  weight  or  mass  are  also, 
of  necessity,  astronomical  and  not  terrestrial.  The  mass 
of  a  body  may  be  expressed  in  terms  of  that  of  the  sun 
or  of  the  earth,  but  never  in  kilograms  or  tons,  unless  in 
popular  language.  There  are  two  reasons  for  this  coarse. 
One  is  that  in  most  cases  celestial  distances  have  first  to 
be  determined  in  terms  of  some  celestial  unit — the  earth's 
distance  from  the  sun,  for  instance — and  it  is  more  con- 
venient to  retain  this  unit  than  to  adopt  a  new  one.  The 
other  is  that  the  values  of  celestial  distances  in  terms  of 
ordinary  terrestrial  units  are  for  the  most  part  extremely 
uncertain,  while  the  corresponding  values  in  astronomical 
units  are  known  with  great  accuracy. 

An  extreme  instance  of  this  is  afforded  by  the  dimen- 
sions of  the  solar  system.  By  a  long  and  continued  series 
of  astronomical  observations,  investigated  by  means  of 
KEPLER'S  laws  and  the  theory  of  gravitation,  it  is  possible 
to  determine  the  forms  of  the  planetary  orbits,  their 
positions,  and  their  dimensions  in  terms  of  the  earth's 


214  ASTRONOMY. 

mean  distance  from  the  sun  as  the  unit  of  measure,  with 
great  precision.  It  will  be  remembered  that  KEPLER'S 
third  law  enables  us  to  determine  the  mean  distance  of  a 
planet  from  the  sun  when  we  know  its  period  of  revolu- 
tion. Now,  all  the  major  planets,  as  far  out  as  Saturn, 
have  been  observed  through  so  many  revolutions  that  their 
periodic  times  can  be  determined  with  great  exactness — in 
fact  within  a  fraction  of  a  millionth  part  of  their  whole 
amount.  The  more  recently  discovered  planets,  Uranus 
and  Neptune,  will,  in  the  course  of  time,  have  their 
periods  determined  with  equal  precision.  Then,  if  we 
square  the  periods  expressed  in  years  and  decimals  of  a 
year,  and  extract  the  cube  root  of  this  square,  we  have  the 
mean  distance  of  the  planet  with  the  same  order  of  pre- 
cision. This  distance  is  to  be  corrected  slightly  in  conse- 
quence of  the  attractions  of  the  planets  on  each  other,  but 
these  corrections  also  are  known  with  great  exactness. 
Again,  the  eccentricities  of  the  orbits  are  exactly  deter- 
mined by  careful  observations  of  the  positions  of  the  plan- 
ets during  successive  revolutions.  Thus  we  are  enabled  to 
make  a  map  of  the  planetary  orbits  which  shall  be  so  ex- 
act that  the  error  would  entirely  elude  the  most  care  hi  I 
scrutiny,  though  the  map  itself  should  be  many  yards  in 
extent. 

On  the  scale  of  this  same  map  we  could  lay  dowr»  the 
magnitudes  of  the  planets  with  as  much  precision  as  our 
instruments  can  measure  their  angular  semi-diameters. 
Thus  we  know  that  the  mean  diameter  of  the  sun,  as  seen 
from  the  earth,  is  32',  hence  we  deduce  from  formulae 
given  in  connection  with  parallax  (Chapter  I.,  §  9),  that 
the  diameter  of  the  sun  is  •  0093083  of  the  distance  of  the 
sun  from  the  earth.  We  can  therefore,  on  our  supposed 
map  of  the  solar  system,  lay  down  the  sun  in  its  true  size, 
according  to  the  scale  of  the  map,  from  data  given  directly 
by  observation.  In  the  same  way  we  can  do  this  for  each 
of  the  planets,  the  earth  and  moon  excepted.  There  is 
no  immediate  and  direct  way  of  finding  how  large  the 


CELESTIAL   MEASURES.  215 

earth  or  moon,  would  look  from  a  planet,  hence  the  ex- 
ception. 

But  without  further  special  research  into  this  subject, 
we  shall  know  nothing  about  the  scale  of  our  map.  It  is 
clear  that  in  order  to  fix  the  distances  or  the  magnitudes 
of  the  planets  according  to  any  terrestrial  standard,  we 
must  know  this  scale.  Of  course  if  we  can  learn  either 
the  distance  or  magnitude  of  any  one  of  the  planets  laid 
down  on  the  map,  in  miles  or  in  semi-diameters  of  the 
earth,  we  shall  be  able  at  once  to  find  the  scale.  But  this 
process  is  so  difficult  that  the  general  custom  of  astrono- 
mers is  not  to  attempt  to  use  an  exact  scale,  but  to  employ 
the  mean  distance  of  the  sun  from  the  earth  as  the  unit  in 
celestial  measurements.  Thus,  in  astronomical  language, 
we  say  that  the  distance  of  Mercury  from  the  sun  is 
0-387,  that  of  Venus  0-723,  that  of  Mars  1-523,  that 
of  Saturn  9  •  539,  and  so  on.  But  this  gives  us  no  in- 
formation respecting  the  distances  and  magnitudes  in  terms 
of  terrestrial  measures.  The  unknown  quantities  of  our 
map  are  the  magnitude  of  the  earth  on  the  scale  of  the 
map,  and  its  distance  from  the  sun  in  terrestrial  units  of 
length.  Could  we  only  take  up  a  point  of  observation 
from  the  sun  or  a  planet,  and  determine  exactly  the  angu- 
lar magnitude  of  the  earth  as  seen  from  that  point,  we 
should  be  able  to  lay  down  the  earth  of  our  map  in  its  cor- 
rect size.  Then  since  we  already  know  the  size  of  the 
earth  in  terrestrial  units,  we  should  be  able  to  find  the 
scale  of  our  map,  and  thence  the  dimensions  of  the  whole 
system  in  terms  of  those  units. 

It  will  be  seen  that  what  the  astronomer  really  wants  is 
not  so  much  the  dimensions  of  the  solar  system  in  miles  as 
to  express  the  size  of  the  earth  in  celestial  measures. 
These,  however,  amount  to  the  same  thing,  because  hav- 
ing one,  the  other  can  be  readily  deduced  from  the  known 
magnitude  of  the  earth  in  terrestrial  measures. 

The  magnitude  of  the  earth  is  not  the  only  unknown 
quantity  on  our  map.  From  KEPLER'S  laws  we  can  de- 


216  ASTRONOMY. 

termine  nothing  respecting  the  distance  of  the  moon  from 
the  earth,  because  unless  a  change  is  made  in  the  units  of 
time  and  space,  they  apply  only  to  bodies  moving  around 
the  sun.  We  must  therefore  determine  the  distance  of 
the  moon  as  well  as  that  of  the  sun  to  be  able  to  complete 
our  map  on  a  known  scale  of  measurement. 

§   2.    MEASURES  OP  THE  SOLAR  PARALLAX. 

The  problem  of  distances  in  the  solar  system  is  reduced 
by  the  preceding  considerations  to  measuring  the  distances 
of  the  sun  and  moon  in  terms  of  the  earth's  radius.  The 
most  direct  method  of  doing  this  is  by  determining  their 
respective  parallaxes,  which  we  have  shown  to  be  the  same 
as  the  earth's  angular  semi-diameter  as  seen  from  them. 
In  the  case  of  the  sun,  the  required  parallax  can  be  de- 
termined as  readily  by  measuring  the  parallaxes  of  any 
of  the  planets  as  by  measuring  that  of  the  sun,  because 
any  one  measured  distance  on  the  map  will  give  us  the 
scale  of  our  map.  Now,  the  planets  Venus  and  Mars  oc- 
casionally come  much  nearer  the  earth  than  the  sun  ever 
does,  and  their  parallaxes  also  admit  of  more  exact  meas- 
urement. The  parallax  of  the  sun  is  therefore  determined 
not  by  observations  on  the  sun  itself,  but  on  these  two 
^lanets.  Three  methods  of  finding  the  sun's  parallax  in 

is  way  have  been  applied.     They  are  : 

(1.)  Observations  of  Venus  in  transit  across  the  sun. 

(2.)  Observations  of  the  declination  of  Mars  from 
widely  separated  stations  on  the  earth's  surface. 

(3.)  Observations  of  the  right  ascension  of  Mars,  near 
the  times  of  its  rising  and  setting,  at  a  single  station. 

Solar  Parallax  from  Transits  of  Venus. —  The  general 
principles  of  the  method  of  determining  the  parallax  of  a 
planet  by  simultaneous  observations  at  distant  stations 
will  be  seen  by  referring  to  Fig.  18,  p.  49.  If  two  ob- 
servers, situated  at  8f  and  /£",  make  a  simultaneous  ob- 
servation of  the  direction  of  the  body  P,  it  is  evident 


TRANSITS  OF  VENUS.  217 

that  the  solution  of  a  plane  triangle  will  give  the  distance 
of  P  from  each  station.  In  practice,  however,  it  would 
he  impracticable  to  make  simultaneous  observations  at 
distant  stations,  and  as  the  planet  is  continually  in  motion, 
the  problem  is  a  much  more  complex  one  than  that  of 
simply  solving  a  triangle.  The  actual  solution  is  effected 
by  a  process  which  is  algebraic  rather  than  geometrical, 
but  we  may  briefly  describe  the  geometrical  nature  of  the 
problem. 

Considering  the  problem  as  a  geometrical  one,  it  is  evi- 
dent that,  owing  to  the  parallax  of  Venus  being  nearly  four 
times  as  great  as  that  of  the  sun,  its  path  across  the  sun's 
disk  will  be  different  when  viewed  from  different  points  of 
the  earth's  surface.  The  further  south  we  go,  the  further 
north  the  planet  will  seem  to  be  on  the  sun's  disk.  The 
change  will  be  determined  by  the  difference  between  the 
parallax  of  Venus  and  that  of  the  sun,  and  this  makes  the 
geometrical  explanation  less  simple  than  in  the  case  of  a 
determination  into  which  only  one  parallax  enters.  It 
will  be  sufficient  if  the  reader  sees  that  when  we  know  the 
relation  between  the  two  parallaxes — when,  for  instance, 
we  know  that  the  parallax  of  Venus  is  3-78  times  that  of 
the  sun — the  observed  displacement  of  Venus  on  the  sun's 
disk  will  give  us  both  parallaxes.  The  "  relative  paral- 
lax," as  it  is  called,  will  be  2-78  times  the  sun's  parallax." 
mid  it  is  on  this  alone  that  the  displacement  depends. 

The  algebraic  process,  which  is  that  actually  employed  in  the 
solution  of  astronomical  problems  of  this  class,  is  as  follows  : 

Each  observer  is  supposed  to  know  his  longitude  and  lati- 
tude, and  to  have  made  one  or  more  observations  of  the  angular 
distance  of  the  centre  of  the  planet  from  the  centre  of  the  sun. 

To  work  up  the  observations,  the  investigator  must  have  an 
ephemeris  of  Venus  and  of  the  sun — that  is,  a  table  giving 
the  right  ascension  and  declination  of  each  body  from  hour  to  hour 
as  calculated  from  the  best  astronomical  data.  The  ephemeris  can 
never  be  considered  absolutely  correct,  but  its  error  may  be  as- 
sumed as  constant  for  an  entire  day  or  more.  By  means  of  it,  the 
right  ascension  and  declination  of  the  planet  and  of  the  sun,  as  seen 
from  the  centre  of  the  earth,  may  be  computed  at  any  time. 

Each  observer  reduces  the  moments  of  his  observations  to  Green- 


218  ASTRONOMY. 

wich  mean  time,  or  the  mean  time  of  any  other  meridian.  Lcl 
those  mean  times  for  the  observer  Si  be  called  rl\,  T*,  T3,  etc. 
Suppose  that  at  these  mean  times  he  has  observed  the  distances  of 
the  centre  of  Venus  from  that  of  the  sun  to  be  DI,  JD2,  Da,  etc. 
The  corresponding  geocentric  distances  are  then  computed  froir 
the  ephemeris  for  these  same  times,  Ti,  T^  T3,  etc.  If  the  ephem- 
eris  and  the  observations  were  perfectly  correct,  and  if  there  were 
no  parallax,  these  calculated  distances  would  come  out  the  same  as 
the  observed  ones.  But  this  is  never  the  case.  It  is  therefore 
necessary  to  calculate  what  effect  a  change  in  the  right  ascension, 
declination,  and  parallax  of  the  sun  and  Venus  will  have  upon  the 
calculated  distance.  In  this  operation  these  changes  are  considered 
as  infinitely  small,  and  the  process  used  is  that  of  differentiation. 
Let  us  put  : 

a,  8,  TT,  the  right  ascension,  declination,  and  parallax  of    Venus. 

a',  <V,  ?r',  the  same  quantities  for  the  sun. 

A  or,  A  <f,  A  a',  A  d',  the  corrections  necessary  to  the  values  of  the 
quantities  :  a,  <5,  a',  and  8  in  the  ephemeris. 

dij  di,  d3,  etc.,  the  calculated  geocentric  distances  of  Venus  from 
the  sun's  centre. 

Then,  the  corrected  calculated  distances,  which  we  shall  call 
D'i,  J9'2,  D'3,  etc.,  will  be  expressed  in  equations  of  the  form  : 

di  +  «i  A  a  +  a\  A  a'  +  &i  A  8  +  l\  A  6'  +  Ci  TT  +  c'i  IT'  =  D\  ; 
rfa  +  «2  A  a  +  «'2  A  a'  +  Z>2  A  6  +  J'4  A  6'  +  c2  TT  +  c'a  TT'  =  D\. 

In  these  equations  dtj  d*,  etc.,  and  the  coefficients,  «i,  «i,  «2,  etc., 
to  c'2,  are  all  known  quantities,  being  the  direct  results  of  calcula- 
tion, while  A  a,  A  or,  Ad,  and  A<5  are  unknown  corrections  to  the 
ephemeris,  and  TT  and  TT'  are  the  parallaxes  of  Venus  and  the  sun, 
also  unknown.  D'i,  -ZXa,  etc.,  are  therefore  also  to  be  regarded  as 
unknown. 

But  when  all  corrections  are  allowed  for,  these  corrected  calcu- 
lated distances  D'i,  D'2,  etc.,  ought  to  be  the  same  as  the  observed 
distances  D'l,  D'2,  etc.,  which  are  known  quantities,  being  the  direct 
result  of  observations.  So  if  we  put  Dl  for  Z>'i,  etc.,  and  transpose 
di  to  the  other  side  of  the  equation,  and  perform  the  same  process 
on  the  other  equations,  we  shall  have  : 

ai  A  a  +  a\  A  a'  +  &,  A  8  +  &',  A  6'  +  d  TT  +  c\  TT'  =  Di  —  di 
«3  A  a  +  «'2  A  a'  -f  52  A  d  -f  J'2  A  <$'  +  c2  TT  +  c'2  TT'  =  Di  —  d^  etc. 
These  equations  admit  of  being  much  simplified.     If  we  suppose 
the  right  ascensions  of  the  sun  and  Venus  changed  by  the  same  amount 
—that  is,  if  we  suppose  A«'  =  A  a,  it  is  evident  that  their  distances 
will  remain  substantially  unaltered.    In  order  that  this  may  be  true 
in  the  equations,  we  must  have 


because  the  real  change  will  be,  in  the  case  supposed, 
«i  A  a  -f  a'i  A  a  =  (0,1  +  a'i)  A  a  =  0. 


TRANSITS  OF  VENUS.  219 

In  the  same  way,  we  must  have  very  nearly, 

lt'\  =  —  bi  ;  c'i  =  —  Ci. 

Then  if  we  substitute  these  values  of  the  accented  coefficients,  the 
first  equation  will  be  : 

«i  (A  a  —  A  or')  +  &,  (A  (5  —  A  d')  -f  c,  (TT  —  TT')  =  Z>,  —  ^L 
If  we  put  for  brevity, 

a;  =  A  a-  —  A  n:'  ;  y  =  A  cJ  —  A  <T, 
the  equations  will  become  : 

«i  «  +  £1  y  +  Ci  (TT  —  Tr7)  =  Z>,  —  d, 


The  parallaxes  of  the  sun  and  Venus,  TT'  and  TT,  are  inversely  as  the 
distances  of  the  respective  bodies  from  the  earth.  During  the  tran- 
sit of  December,  1874,  these  distances  were: 

Distance  of  sun,       0-9847, 
"         "     Venus,  0-2644. 

So,  if  we  put  7T0  for  the  parallax  at  distance  1,  we  shall  have  : 

Actual  parallax  of  the  sun,  TT'  =  —   —  =  1-0155  TTO. 

u  * 


Actual  parallax  of  Venus,     K  =  _^  —  =  3-7822  TTO  ; 

0  •  2644 
whence 

TT  —  7r'=r  2-7667^0. 

Substituting  this  value  in  our  equations,  they  will  become  : 

«i  x  +  &,  y  +  2  •  7667  c,  ?r0  =  Dl  —  di 

a*  x  +  62  y  -f  2  •  7667  c2  TTO  =  D2  —  d2,  etc. 

All  the  corresponding  equations  being  formed  in  this  way,  from 
the  observations  at  the  various  stations,  their  solution  will  give  the 
values  of  the  three  unknown  quantities,  *,  y,  and  TTO.  The  value  of 
TTo  will  be  the  parallax  corresponding  to  the  astronomical  unit  — 
that  is,  the  angular  semi-diameter  of  the  earth  seen  at  the  mean 
distance  from  the  sun. 

When  many  observations  are  made,  we  have  more  equations  than 
there  are  unknown  quantities  to  be  determined.  If  all  the  equations 
were  mathematically  correct,  we  should  not  need  them  all,  and  could 
reject  any  of  the  surplus  ones  without  affecting  the  result.  But 
since  each  equation  is  necessarily  affected  with  errors  of  observa- 
tion, the  problem  presented  to  us  is  to  obtain  the  most  probable 
values  of  the  unknown  quantities  from  the  combination  of  all  the 
equations.  These  values  are  those  which  render  the  sum  of  the 
quares  of  the  outstanding  errors  of  observations  (or,  rather,  of 


220  ASTRONOMY. 

the  outstanding  differences  between  the  observed  quantities  and 
the  computed  quantities)  a  minimum.  For  instance,  suppose  that 
we  substitute  in  the  equation 

«i  x  +  Z>i  y  +  2  •  7767  Ci  TTO  =  Dl  —  d, 

any  assumed  values  of  «,  y,  and  TTO.  In  general  the  equation  will 
not  be  satisfied,  but  there  will  remain  a  small  difference  between 
the  two  members,  which  we  may  call  A,.  Let  us  call  A2  the  differ- 
ence obtained  in  the  same  way  from  the  second  equation,  A3  from 
the  third,  and  so  on,  and  let  us  put  8  for  the. sum  of  the  squares  of 
these  quantities,  so  that 

S=  A2,  +  A22  +  A23  +  etc. 

Then,  for  each  system  of  values  of  v,  y,  and  TTO,  which  we  choose  to 
assume,  there  will  be  a  corresponding  value  of  /S,  and  the  most 
probable  system  of  values  will  be  that  which  makes  8  the  least. 

The  method  by  which  this  result  is  reached  is  called  the  method 
of  least  squares,  and  is  developed  in  works  on  astronomical  compu- 
tations. 

Measurements  of  the  Parallax  of  Mars.— This  parallax  may 
be  determined  from  observations  in  two  ways.  In  that  usually 
adopted  there  are  two  observers  or  sets  of  observers,  one  in  the 
northern  and  the  other  in  the  southern  hemisphere,  each  of  whom 
determines  the  declination  of  the  planet  from  day  to  day  at  the 
moment  of  transit  over  his  meridian.  These  declinations  will  be 
different  by  the  whole  amount  of  parallactic  difference  between  the 
two  stations,  or  by  the  angle  8'  PS"  in  Fig.  18,  p.  49.  The  observa- 
tions are  continued  through  the  period  when  Mars  is  nearest  the  earth, 
generally  about  a  couple  of  months.  Any  opposition  of  the  planet 
may  be  chosen  for  this  purpose,  but  the  most  favorable  ones  are 
those  when  the  planet  is  nearest  its  perihelion.  Should  the  planet 
be  exactly  at  its  perihelion  at  the  time  of  opposition,  its  distance 
from  the  earth  would  be  only  about  0  •  37,  while  at  aphelion  it  would 
be  0-68.  This  great  difference  is  owing  to  the  considerable  eccen- 
tricity of  the  orbit  of  Mars,  as  can  be  seen  by  studying  Fig.  48, 
p.  115,  which  gives  a  plan  of  most  of  the  orbits  of  the  larger  planets. 
The  favorable  oppositions  occur  at  intervals  of  15  or  17  years.  One 
was  that  of  1862,  which  gave  almost  the  first  conclusive  evidence  that 
the  old  parallax  of  the  sun  found  by  ENCKE  was  too  small.  This 
parallax  was  8"»577,  and  the  corresponding  distance  of  the  sun  was 
95|-  millions  of  miles.  The  observations  of  1862  seemed  to  show 
that  this  parallax  must  be  increased  by  about  one  thirtieth  part,  and 
the  distance  diminished  in  about  the  same  ratio.  But  the  most  recent 
results  make  it  probable  that  the  change  should  not  be  quite  so 
great  as  this. 

An  extremely  favorable  opposition,  in  respect  of  distance,  was  that 
of  September  5th,  1877,  which  occurred  15  days  after  Mars  passed 
its  perihelion.  On  September  3d  its  distance  from  the  earth  was 
only  0-377— less  than  it  had  been  at  any  time  since  August,  1845. 


PARALLAX  OF  MARS.  221 

Parallax  of  Mars  in  Right  Ascension.— Another  method 
of  measuring  the  parallax  of  Mars  is  founded  on  principles  entirely 
different  from  those  we  have  hitherto  considered.  In  the  latter, 
observations  have  to  be  made  by  two  observers  in  opposite  hemi- 
spheres of  the  earth.  But  an  observer  at  any  point  on  the  earth's 
surface  is  carried  around  on  a  circle  of  latitude  every  day  by  the 
diurnal  motion  of  the  earth.  In  consequence  of  this  motion,  there 
must  be  a  corresponding  apparent  motion  of  each  of  the  planets  in 
an  opposite  direction.  In  other  words,  the  parallax  of  the  planet 
must  be  different  at  different  times  of  the  day.  This  diurnal 
change  in  the  direction  of  the  planet  admits  of  being  measured  in 
the  following  way  :  The  effect  of  parallax  is  always  to  make  a 
heavenly  body  appear  nearer  the  horizon  than  it  would  appear  as  seen 
from  the  centre  of  the  earth.  This  will  be  obvious  if  we  reflect 
that  an  observer  moving  rapidly  from  the  centre  of  the  earth  to  its 
circumference,  and  keeping  his  eye  fixed  upon  a  planet,  would  see 
the  planet  appear  to  move  in  an  opposite  direction — that  is,  down- 
ward relative  to  the  point  of  the  earth's  surface  which  he  aimed  at. 
Hence  a  planet  rising  in  the  east  will  rise  later  in  consequence  of 
parallax,  and  will  set  earlier.  Of  course  the  rising  and  setting 
cannot  be  observed  with  sufficient  accuracy  for  the  purpose  of 
parallax,  but,  since  a  fixed  star  has  no  parallax,  the  position  of 
the  planet  relative  to  the  stars  in  its  neighborhood  will  change 
during  the  interval  between  the  rising  and  setting  of  the  planet. 
The  observer  therefore  determines  the  positon  of  Mars  relative 
to  the  stars  surrounding  him  shortly  after  he  rises  and  again 
shortly  before  he  sets.  The  observations  are  repeated  night 
after  night  as  often  as  possible.  Between  each  pair  of  east  and 
west  observations  the  planet  will  of  course  change  its  position 
among  the  stars  in  consequence  of  the  orbital  motions  of  the 
earth  and  planet,  but  these  motions  can  be  calculated  and  allowed 
for,  and  the  changes  still  outstanding  will  then  be  due  to  parallax. 

The  most  favorable  regions  for  an  observer  to  determine  the  par- 
allax in  this  way  are  those  near  the  earth's  equator,  because  he  is 
there  carried  around  on  the  largest  circle.  If  he  is  nearer  the  poles 
than  the  equator,  the  circle  will  be  so  small  that  the  parallax  will  be 
hardly  worth  determining,  while  at  the  poles  there  will  be  no  par- 
allactic  change  at  all  of  the  kind  just  described. 

Applications  of  this  method  have  not  been  very  numerous, 
although  it  was  suggested  by  FLAMSTEED  nearly  two  centuries  ago. 
The  latest  and  most  successful  trial  of  it  was  made  by  Mr.  DAVID  GILL 
of  England  during  the  opposition  of  Mars  in  1877  above  described. 
The  point  of  observation  chosen  by  him  was  the  island  of  Ascen- 
sion, west  of  Africa  and  near  the  equator.  His  measures  indicate 
a  considerable  reduction  in  the  recently  received  values  of  the  solar 
parallax,  and  an  increase  in  the  distance  of  the  sun,  making  the 
latter  come  somewhat  nearer  to  the  old  value. 

Accuracy  of  the  Determinations  of  Solar  Parallax.-- 
The   parallax  of  Mars  at  opposition  is  rarely  more  than 


222  ASTRONOMY. 

20",  and  the  relative  parallax *of  Venus  and  the  sun  at  the 
time  of  the  transit  is  less  than  24".  These  quantities  are 
so  small  as  to  almost  elude  very  precise  measurement  ;  it 
is  hardly  possible  by  any  one  set  of  measures  of  parallax 
to  determine  the  latter  without  an  uncertainty  of  -g-J^  of  its 
whole  amount.  In  the  distance  of  the  sun  this  corre- 
sponds to  an  uncertainty  of  nearly  half  a  million  of  miles. 
Astronomers  have  therefore  sought  for  other  methods  of 
determining  the  sun's  distance.  Although  some  of  these 
may  be  a  little  more  certain  than  measures  of  parallax,  there 
is  none  by  which  the  distance  of  the  sun  can  be  determined 
with  any  approximation  to  the  accuracy  which  character- 
izes other  celestial  measures. 

Other  Methods  of  Determining  Solar  Parallax. —  A 
very  interesting  and  probably  the  most  accurate  method 
of  measuring  the  sun's  distance  is  by  using  light  as  a  mes- 
senger between  the  sun  and  the  earth.  We  shall  hereafter 
see,  in  the  chapter  on  aberration,  that  the  time  required  for 
light  to  pass  from  the  sun  to  the  earth  is  known  with  con- 
siderable exactness,  being  very  nearly  498  seconds.  If 
then  we  can  determine  experimentally  how  many  miles  or 
kilometres  light  moves  in  a  second,  we  shall  at  once  have 
the  distance  of  the  sun  by  multiplying  that  quantity  by 
498.  But  the  velocity  of  light  is  about  300,000  kilometres 
per  second.  This  distance  would  reach  about  eight  times 
around  the  earth.  It  is  rarely  possible  that  two  points  on 
the  earth's  surface  more  than  a  hundred  kilometres  apart 
are  visible  from  each  other,  and  distinct  vision  at  distances 
of  more  than  twenty  kilometres  is  rare.  Hence  to  deter- 
mine experimentally  the  time  required  for  light  to  pass 
between  two  terrestrial  stations  requires  the  measurement  of 
an  interval  of  time,  which  even  under  the  most  favorable 
cases  can  be  only  a  fraction  of  a  thousandth  of  a  second. 
Methods  of  doing  it,  however,  have  been  devised  and  ex- 
ecuted by  the  French  physicists,  FIZEAU,  FOUOAULT,  and 
CORNU,  and  quite  recently  by  Ensign  MICHELSON  at  the 
U.  S.  Naval  Academy,  Annapolis.  From  the  experiments 


SOLAR  PARALLAX.  223 

of  the  latter,  which  are  probably  the  most  accurate,  the 
velocity  of  light  would  seem  to  be  about  299,900  kilome- 
tres per  second.  Multiplying  this  by  498,  we  obtain  149,- 
350,000  kilometres  for  the  distance  of  the  sun.  The  time 
required  for  light  to  pass  from  the  sun  to  the  earth  is  still 
uncertain  by  nearly  a  second,  but  this  value  of  the  sun's 
distance  is  probably  the  best  yet  obtained.  The  corre- 
sponding value  of  the  sun's  parallax  is  8" '81. 

Yet  other  methods  of  determining  the  sun's  distance 
are  given  by  the  theory  of  gravitation.  The  best  known 
of  these  depends  upon  the  determination  of  the  parallactic 
inequality  of  the  moon.  It  is  found  by  mathematical  in- 
vestigation that  the  motion  of  the  moon  is  subjected  to 
several  inequalities,  having  the  sun's  horizontal  parallax 
as  a  factor.  In  consequence  of  the  largest  of  these  in- 
equalities, the  moon  is  about  two  minutes  behind  its  mean 
place  near  the  first  quarter,  and  as  far  in  advance  at  the 
last  quarter.  If  the  position  of  the  moon  could  be  deter- 
mined by  observation  with  the  same  exactness  that  the  po- 
sition of  a  star  or  planet  can,  this  would  probably  afford 
the  most  accurate  method  of  determining  the  solar  par- 
allax. But  an  observation  of  the  moon  has  to  be  made, 
not  upon  its  centre,  but  upon  its  limb  or  circumference. 
Only  the  limb  nearest  the  sun  is  visible,  the  other  one 
being  unilluminated,  and  thus  the  illuminated  limb  on 
which  the  observation  is  to  be  made  is  different  at  the  first 
and  third  quarter.  These  conditions  induce  an  uncertain- 
ty in  the  comparison  of  observations  made  at  the  two 
quarters  which  cannot  be  entirely  overcome,  and  therefore 
leave  a  doubt  respecting  the  correctness  of  the  result. 

Brief  History  of  Determinations  of  the  Solar  Parallax. 
—The  distance  of  the  sun  must  at  all  times  have  been  one 
of  the  most  interesting  scientific  problems  presented  to  the 
human  mind.  The  first  known  attempt  to  effect  a  solu- 
tion of  the  problem  was  made  by  ARISTARCHUS,  who  flour- 
ished in  the  third  century  before  CHRIST.  It  was  founded 
on  the  principle  that  the  time  of  the  moon's  first  quarter 


224  ASTRONOMY. 

will  vary  with  the  ratio  between  the  distance  of  the  moon 
and  sun,  which  may  be  shown  as  follows.  In  Fig.  68 
let  .#  represent  the  earth,  M  the  moon,  and  8  the  sun. 
Since  the  sun  always  illuminates  one  half  of  the  lunar 
globe,  it  is  evident  that  when  one  half  of  the  moon's  disk 
appears  illuminated,  the  triangle  EMS  must  be  right- 
angled  at  M.  The  angle  M  JE  S  can  be  determined  by 
measurement,  being  equal  to  the  angular  distance  between 
the  sun  and  the  moon.  Having  two  of  the  angles,  the 
third  can  be  determined,  because  the  sum  of  the  three 
must  make  two  right  angles.  Thence  we  shall  have  the 
ratio  between  E M,  the  distance  of  the  moon,  and  ES* 
the  distance  of  the  sun,  by  a  trigonometrical  computation. 


FIG.  70. 

Then  knowing  the  distance  of  the  moon,  which  can  be 
determined  with  comparative  ease,  we  have  the  distance  of 
the  sun  by  multiplying  by  this  ratio.  ARISTARCIIUS  con- 
cluded, from  his  supposed  measures,  that  the  angle  M  £8 
was  three  degrees  less  than  a  right  angle.  We  should 

then  have  -^-r/  =  si11  3°  =  -fa   very    nearly.       It   would 

M*  -M. 

follow  from  this  that  the  sun  was  19  times  the  distance 
of  the  moon.  We  now  know  that  this  result  is  entirely 
wrong,  and  that  it  is  impossible  to  determine  the  time 
when  the  moon  is  exactly  half  illuminated  with  any  ap- 
proach to  the  accuracy  necessary  in  the  solution  of  the 
problem.  In  fact,  the  greatest  angular  distance  of  the 


SOLAR  PARALLAX.  225 

earth  and  moon,  as  seen  from  the  sun — that  is,  the  angle 
ES  M — is  only  about  one  quarter  the  angular  diameter  of 
the  moon  as  seen  from  the  earth. 

The  second  attempt  to  determine  the  distance  of  the 
sun  is  mentioned  by  PTOLEMY,  though  HIPPARCHUS  may  be 
the  real  inventor  of  it.  It  is  founded  on  a  somewhat  com- 
plex geometrical  construction  of  a  total  eclipse  of  the 
moon.  It  is  only  necessary  to  state  the  result,  which 
was,  that  the  sun  was  situated  at  the  distance  of  1210  radii 
of  the  earth.  This  result,  like  the  former,  was  due  only 
to  errors  of  observation.  So  far  as  all  the  methods  known 
at  the  time  could  show,  the  real  distance  of  the  sun  ap- 
peared to  be  infinite,  nevertheless  PTOLEMY'S  result  was 
received  without  question  for  fourteen  centuries. 

When  the  telescope  was  invented,  and  more  accurate 
observations  became  possible,  it  was  found  that  the  sun's 
distance  must  be  greater  and  its  parallax  smaller  than 
PTOLEMY  had  supposed,  but  it  was  still  impossible  to  give 
any  measure  of  the  parallax.  All  that  could  be  said  was 
that  it  was  less  than  the  smallest  quantity  that  could  be  de- 
cided on  by  measurement.  The  first  approximation  to  the 
true  value  was  made  by  HORROX  of  England,  and  after- 
ward by  HUYGHENS  of  Holland.  It  was  not  founded  on 
any  attempt  to  measure  the  parallax  directly,  but  on  an 
estimate  of  the  probable  magnitude  of  the  earth  on  the 
scale  of  the  solar  system.  The  magnitude  of  the  planets 
on  this  scale  being  known  by  measurement  of  their  appar- 
ent angular  diameters  as  seen  from  the  earth,  the  solar 
parallax  may  be  found  when  we  know  the  ratio  between 
the  diameter  of  the  earth  and  that  of  any  planet  whose 
angular  diameter  has  been  measured.  Now,  it  was  sup- 
posed by  the  two  astronomers  we  have  mentioned  that 
the  earth  was  probably  of  the  same  order  of  magnitude 
with  the  other  planets. 

HORROX  had  a  theory,  which  we  now  know  to  be  erro- 
neous, that  the  diameters  of  the  planets  were  proportional 
to  their  distances  from  the  sun — in  other  words,  that  all 


226  ASTRONOMY. 

the  planets  would  appear  of  fhe  same  diameter  when  seen 
from  the  sun.  This  diameter  he  estimated  at  28",  from 
which  it  followed  that  the  solar  parallax  was  14".  HUYGHENS 
assumed  that  the  actual  magnitude  of  the  earth  was  mid- 
way between  those  of  the  two  planets  Venus  and  Mars  on 
each  side  of  it ;  he  thus  obtained  a  result  remarkably  near 
the  truth.  It  is  true  that  in  reality  the  earth  is  a  little 
larger  than  either  Venus  or  Mars,  but  the  imperfect  tel- 
escopes of  that  time  showed  the  planets  larger  than  they 
really  were,  so  that  the  mean  diameter  of  the  enlarged 
planets,  as  seen  in  the  telescope  of  HUYGHENS,  was  such  as 
to  correspond  very  nearly  to  the  diameter  of  the  earth. 

The  first  really  successful  measure  of  the  parallax 
of  a  planet  was  made  upon  Mars  during  the  opposition  of 
1672,  by  the  first  of  the  two  methods  already  described. 
An  expedition  was  sent  to  the  colony  of  Cayenne  to  ob- 
serve the  declination  of  the  planet  from  night  to  night, 
while  corresponding  observations  were  made  at  the  Paris 
Observatory.  From  a  discussion  of  these  observations, 
CASSINI  obtained  a  solar  parallax  of  9"  •  5,  which  is  within 
a  second  of  the  truth.  The  next  steps  forward  were  made 
by  the  transits  of  Venus  in  1761  and  1769.  The  leading 
civilized  nations  caused  observations  on  these  transits  to  be 
made  at  various  points  on  the  globe.  The  method  used 
was  very  simple,  consisting  in  the  determination  of  the 
times  at  which  Venus  entered  upon  the  sun's  disk  and  left 
it  again.  The  absolute  times  of  ingress  and  egress,  as  seen 
from  different  points  of  the  globe,  might  differ  by  20 
minutes  or  more  on  account  of  parallax.  The  results, 
however,  were  found  to  be  discordant.  It  was  not  until 
more  than  half  a  century  had  elapsed  that  the  observations 
were  all  carefully  calculated  by  ENCKE  of  Germany,  who 
concluded  that  the  parallax  of  the  sun  was  8"  •  857,  and  the 
distance  95  millions  of  miles. 

In  1854  it  began  to  be  suspected  that  ENCKE'S  value  of 
the  parallax  was  much  too  small,  and  great  labor  was  now 
devoted  to  a  solution  of  the  problem.  HANSEN,  from  the 


MASSES  OF  THE  SUN  AND  EARTH.  227 

parallactic  inequality  of  the  moon,  first  found  the  parallax 
of  the  sun  to  be  8"  -  97,  a  quantity  which  he  afterward  re- 
duced to  8* -916.  This  result  seemed  to  be  confirmed  by 
other  observations,  especially  those  of  Mars  during  the 
opposition  of  1862.  It  was  therefore  concluded  that  the 
sun's  parallax  was  probably  between  8" -90  and  9" -00. 
Subsequent  researches  have,  however,  been  diminishing 
this  Aralue.  In  1867,  from  a  discussion  on  all  the  data 
which  were  considered  of  value,  it  was  concluded  by  one 
of  the  writers  that  the  most  probable  parallax  was  8*  •  848. 
The  measures  of  the  velocity  of  light  made  by  MICHELSON 
in  1878  reduce  this  value  to  8"»81,  and  it  is  now  doubtful 
whether  the  true  value  is  any  larger  than  this. 

The  observations  of  the  transit  of  Venus  in  1874  have 
not  been  completely  discussed  at  the  time  of  writing  these 
pages.  When  this  is  done  some  further  light  may  be 
thrown  upon  the  question.  It  is,  however,  to  the  deter- 
mination of  the  velocity  of  light  that  we  are  to  look  for 
the  best  result.  All  we  can  say  at  present  is  that  the  so- 
lar parallax  is  probably  between  8"  •  79  and  8"  •  83,  or,  if 
outside  these  limits,  that  it  can  be  very  little  outside. 


§   3.    RELATIVE  MASSES  OP  THE  SUN  AND 
PLANETS. 

Tn  estimating  celestial  masses  as  well  as  distances,  it  is  necessary 
to  use  what  we  may  call  celestial  units — that  is,  to  take  the  mass  of 
some  celestial  body  as  a  unit,  instead  of  any  multiple  of  the  pound 
or  kilogram.  The  reason  of  this  is  that  the  ratios  among  the 
masses  of  the  planetary  system,  or,  which  is  the  same  thing,  the 
mass  of  each  body  in  terms  of  that  of  some  one  body  as  the  unit, 
can  be  determined  independently  of  the  mass  of  any  one  of  them. 
To  express  a  mass  in  kilograms  or  other  terrestrial  units,  it  is  neces- 
sary to  find  the  mass  of  the  earth  in  such  units,  as  already  explained. 
This,  however,  is  not  necessary  for  astronomical  purposes, 'where  only 
the  relative  masses  of  the  several  planets  are  required.  In  estimat- 
ing the  masses  of  the  individual  planets,  that  of  the  sun  is  generally 
taken  as  a  unit.  The  planetary  masses  will  then  all  be  very  small 
fractions. 

Masses  of  the  Earth  and  Sun.— We  shall  first  consider  the 
mass  of  the  earth  because  it  is  connected  by  a  very  curious  relation 
with  the  parallax  of  the  sun.  Knowing  the  latter,  we  can  determine 


228  ASTRONOMY. 

the  mass  of  the  sun  relative  to  the.  earth,  which  is  the  same  thing 
as  determining  the  astronomical  mass  of  the  earth,  that  of  the  sun 
being  unity.  This  may  be  clearly  seen  by  reflecting  that  when  we 
know  the  radius  of  the  earth's  orbit  we  can  determine  how  far  the 
earth  moves  aside  from  a  straight  line  in  one  second  in  consequence 
of  the  attraction  of  the  sun.  This  motion  measures  the  attractive 
force  of  the  sun  at  the  distance  of  the  earth.  Comparing  it  with 
the  attractive  force  of  the  earth,  and  making  allowance  for  the 
difference  of  distances  from  centres  of  the  two  bodies,  we  deter- 
mine the  ratio  between  their  masses. 

The  calculation  in  question  is  made  in  the  most  simple  and  ele- 
mentary manner  as  follows.  Let  us  put  : 

TT,  the  ratio  of  the  circumference  of  a  circle  to  its  diameter  (TT  = 
3-14159  ..  .) 

r,  the  mean  radius  of  the  earth,  or  the  radius  of  a  sphere  having 
the  same  volume  as  the  earth. 

«,  the  mean  distance  of  the  earth  from  the  sun. 

g,  the  force  of  gravity  on  the  earth's  surface  at  a  point  where  the 
radius  is  r  —  that  is,  the  distance  which  a  body  will  fall  in  one 
second. 

</',  the  sun's  attractive  force  at  the  distance  a. 

T,  the  number  of  seconds  in  a  sidereal  year. 

M,  the  mass  of  the  sun. 

m,  the  mass  of  the  earth. 

P,  the  sun's  mean  horizontal  parallax. 

The  force  of  gravity  of  the  sun,  y1  ',  may  be  considered  as  equal  to 
the  so-called  centrifugal  force  of  the  earth,  or  to  the  distance  which 
the  earth  falls  toward  the  sun  in  one  second.  By  the  formula  for 
centrifugal  force  given  in  Chapter  VIII.,  p.  204,  we  have, 


and  by  the  law  of  gravitation, 


whence 


and 


We  have,  in  the  same  way,  for  the  earth, 

_  m 

2'2' 

whence 


MASS  OF  THE  SUN.  229 

Therefore,  for  the  ratio  of  the  masses  of  the  earth  and  sun,  we  have : 

M__^_     o_'_4^      r^     tf 
m~  gT*  '  r*~   T"-  '  </  '  r3 

By  the  formulae  for  parallax  in  Chapter  I.,  §  8,  we  have: 

«3 

r  =  a  sin  P  . '.  —  =  — 


r3  sin3  P 
Therefore 

M_^      r_  I                                   (b 

m~~TT      ~g  sin3  P 

The  quantities  T7,  rand  g  may  be  regarded  as  all  known  with  great 
exactness.  We  see  that  the  mass  of  the  earth,  that  of  the  sun  being 
unity,  is  proportional  to  the  cube  of  the  solar  parallax. 

From  data  already  given,  we  have : 

T=  365  days,  6  hours,  9ni  99;  in  seconds,  T=  31  538  149, 
Mean  radius  of  the  earth  in  metres,*  .  .  r  =  6  370  008, 
Force  of  gravity  in  metres,  .  .  .  .  g  =  9  •  8202, 

while  log  7T2  =  1-59636.     Substituting  these  numbers  in  the  formulae, 
it  may  be  put  in  the  form, 


where  the  quantity  in  brackets  is  the  logarithm  of  the  factor. 

It  will  be  convenient  to  make  two  changes  in  the  parallax  P.  This 
angle  is  so  exceedingly  small  that  we  may  regard  it  as  equal  to  its 
sine.  To  express  it  in  seconds  we  must  multiply  it  by  the  number 
of  seconds  in  the  unit  radius — that  is,  by  206265".  This  will  make 
P  (in  seconds)  =  206265"  sin  P.  Again,  the  standard  to  which  par- 
allaxes are  referred  is  always  the  earth's  equatorial  radius,  which  is 
greater  than  r  by  about  g-fy  of  its  whole  amount.  So,  if  we  put  P" 
for  the  equatorial  horizontal  parallax,  expressed  in  seconds,  we  shall 
have, 

P"  =  (1  +  ^)  206265"  sin  P—  [5-31492]  siiiP, 
whence,  for  sin  P  in  terms  of  P", 


[5-31492J 

*  The  mean  radius  of  the  earth  is  not  the  mean  of  the  polar  and 
equatorial  radii,  but  one  third  the  sum  of  the  polar  radius  and  twice 
the  equatorial  one,  because  we  can  draw  three  such  radii,  each  mak- 
ing a  right  angle  with  the  other  two. 

f  A  number  enclosed  in  brackets  is  frequently  used  to  signify  the 
logarithm  of  a  coefficient  or  divisor  to  be  used. 


230 


ASTRONOMY. 


If  we  substitute  this  value  in  the  expression  for  the  quotient  of 
the  masses,  it  may  be  put  into  either"  of  the  forms  : 

M  =  [8-35493] 
~m  ~        P"3      ' 


P "  =  [2-78498]  f~ Y. 


The  first  formula  gives  the  ratio  of  the  masses  when  the  solar  par- 
allax is  known  ;  the  second,  the  parallax  when  the  ratio  of  the  masses 
is  known.  The  following  table  shows,  for  different  values  of  the 
solar  parallax,  the  corresponding  ratio  of  the  masses,  and  distance  of 
the  sun  in  terrestrial  measures  : 


DISTANCE  OF  THE  SUN. 

SOLAR 
PARALLAX. 
P" 

M 

m 

In  equatorial 
radii  of  the 
earth. 

In  millions  of 
miles. 

In  millions  of 
kilometres. 

8"-  75 

337992 

23573 

93-421 

150-343 

8".  76 

336835 

23546 

93-314 

150-172 

8"  -77 

335684 

23519 

93-208 

150-001 

8"-  78 

334538 

23492 

93-102 

149-830 

8"  -79 

333398 

23466 

92-996 

149-660 

8".  80 

332262 

23439 

92-890 

148-490 

8"-  81 

331132 

23413 

92-785 

149-320 

8".  82 

330007 

23386 

92-680 

149-151 

8".  83 

328887 

23360 

92-575 

148-982 

8"  -84 

327773 

23333 

92-470 

148-814 

8"  -85 

326664 

23307 

92-366 

148-646 

We  have  said  that  the  solar  parallax  is  probably  contained  between 
the  limits  8". 79  and  8".83.  It  is  certainly  hardly  more  than  one  or 
two  hundredths  of  a  second  without  them.  So,  if  we  wish  to  express 
the  constants  relating  to  the  sun  in  round  numbers,  we  may  say  that — 

Its  mass  is  330,000  times  that  of  the  earth. 

Its  distance  in  miles  is  93  millions,  or  perhaps  a  little  less. 

Its  distance  in  kilometres  is  probably  between  149  and  150  mil- 
lions. 

Density  of  the  Sun. — A  remarkable  result  of  the  preceding 
investigation  is  that  the  density  of  the  sun,  relative  to  that  of  the 
earth,  can  be  determined  independently  of  the  mass  or  distance  of 
the  sun  by  measuring  its  apparent  angular  diameter,  and  the  force 
of  gravity  at  the  earth's  surface.  Let  us  put 

/>,  the  density  of  the  sun. 

d,  that  of  the  earth. 

s,  the  sun's  angular  semi-diameter,  as  seen  from  the  earth.  Then, 
continuing  the  notation  already  given,  we  shall  have : 


MASS  OF  THE  SUN.  231 

Linear  radius  of  the  sun  =  a  sin  s. 


Volume  of  the  sun 


47T       -      .      , 

=  —  a3  sin 


(from  the  formula  for  the  volume  of  a  sphere). 

4:TT 

Mass  of  the  sun,    M  =  --^  a3  1)  sin3  *. 

Mass  of  the  earth,  m  =  —  r*d. 
o 

Substituting  these  values  of  M  and  m  in   the  equation  (a),  and 
dividing  out  the  common  factors,  it  will  become 


from  which  we  lirid,  for  the  ratio  of  the  density  of  the  earth  to  that 
of  the  sun, 

d       9  T*     . 


This  equation  solves  the  problem.  But  the  solution  may  be  trans- 
formed in  expression.  We  know  from  the  law  of  falling  bodies  that 
a  heavy  body  will,  in  the  time  t,  fall  through  the  distance  \g  tf. 
Hence  the  factor  g  T*  is  double  the  distance  which  a  body  would  fall 
in  a  sidereal  year,  if  the  force  of  gravity  could  act  upon  it  continu- 
ously with  the  same  intensity  as  at  the  surface  of  the  earth.  Hence 
gT* 
o  will  be  the  number  of  radii  of  the  earth  through  which  the 

body  will  fall  in  a  sidereal  year.     If  we  put  F  for  this  number,  the 
preceding  equation  will  become, 

d       JFsin'a 


We  therefore  have  this  rule  for  finding  the  density  of  the  earth 
relative  to  that  of  the  sun  : 

Find  how  many  radii  of  the  earth  a  heavy  body  would  fall  through 
in  a  sidereal  year  in  virtue  of  the  force  of  gravity  at  the  earth's  sur- 
face. Multiply  this  number  by  the  cube  of  the  sine  of  the  sun's  angular 
semi-diameter,  as  seen  from  the  earth,  and  divide  by  the  numerical 
factor  2  7T2  =  19-7392.  The  quotient  will  be  the  ratio  of  the  density 
of  the  earth  to  that  of  the  sun. 

From  the  numerical  data  already  given,  we  find  : 

Density  of  earth,  that  of  sun  being  unity, 

-  =  3-9208. 


232  ASTRONOMY. 

Density  of  the  sun,  that  of  the  earth  being  unity, 

-  =0-25505. 
a 

These  relations  do  not  give  us  the  actual  density  of  either  body. 
We  have  said  that  the  mean  density  of  the  earth  is  about  5f,  that  of 
water  being  unity.  The  sun  is  therefore  about  40  or  50  per  cent 
denser  than  water. 

Masses  of  the  Planets.— If  we  knew  how  far  a  body  would 
fall  in  one  second  at  the  surface  of  any  other  planet  than  the  earth, 
we  could  determine  its  mass  in  much  the  same  way  as  we  have  de- 
termined that  of  the  earth.  Now  if  the  planet  has  a  satellite  re- 
volving around  it,  we  can  make  this  determination — not  indeed 
directly  on  the  surface  of  the  planet,  but  at  the  distance  of  the  sat- 
ellite, which  will  equally  give  us  the  required  datum.  Indeed  by 
observing  the  periodic  time  of  a  satellite,  and  the  angle  subtended  by 
the  major  axis  of  its  orbit  around  the  planet,  we  have  a  more  direct 
datum  for  determining  the  mass  of  the  planet  than  we  actually  have 
for  determining  that  of  the  earth.  (Of  course  we  here  refer  to  the 
masses  of  the  planets  relative  to  that  of  the  sun  as  unity.)  In  fact 
could  an  astronomer  only  station  himself  on  the  planet  Venus  and 
make  a  series  of  observations  of  the  angular  distance  of  the  moon 
from  the  earth,  he  could  determine  the  mass  of  the  earth,  and 
thence  the  solar  parallax,  with  far  greater  precision  than  we  are  like- 
ly to  know  it  for  centuries  to  come.  Let  us  again  consider  the 
equation  for  M  found  on  page  228  : 


T* 

Here  a  and  T  may  mean  the  mean  distance  and  periodic  time  of 
any  planet,  the  quotient  — -  being  a  constant  by  KEPLER'S  third 

law.  In  the  same  equation  we  may  suppose  a  the  mean  distance  of 
a  satellite  from  its  primary,  and  T  its  time  of  revolution,  and  M  will 
then  represent  the  mass  of  the  planet.  We  shall  have  therefore  for 
the  mass  of  the  planet, 


a'  being  the  mean  distance  of  the  satellite  from  the  planet,  and  T' 
its  time  of  revolution.  Therefore,  for  the  mass  of  the  planet  rel 
ative  to  that  of  the  sun  we  have  : 


ra 


M  ~  a3  T 


Let  us  suppose  a  to  be  the  mean  distance  of  the  planet  from  the 
sun,  in  which  case  T  must  represent  its  time  of  revolution.  Then, 
if  we  put  s  for  the  angle  subtended  by  the  radius  of  the  orbit  of  the 


MASSES  OF  TILE  PLANETS.  233 

satellite,  as  seen  from  the  sun,  we  shall  have,  assuming  the  orbit 
to  be  seen  edgewise, 


sins  =  — . 
a 


If  the  orbit  is  seen  in  a  direction  perpendicular  to  its  plane,  we 
should  have  to  put  tang  s  for  sin  s  in  this  formula,  but  the  angle 
s  is  so  small  that  the  sine  and  tangent  are  almost  the  same.  If  we 
put  T  for  the  ratio  of  the  time  of  revolution  of  the  planet  to  that  of 
the  satellite,  it  will  be  equivalent  to  supposing 


The  equation  for  the  mass  of  the  planet  will  then  become 

m         -2   •  <» 
-—  =  r2  sms  s. 
M 

which  is  the  simplest  form  of  the  usual  formula  for  deducing  the 
mass  of  a  planet  from  the  motion  of  its  satellite.  It  is  true  that  we 
cannot  observe  s  directly,  since  we  cannot  place  ourselves  on  the 
sun,  but  if  we  observe  the  angle  8  from  the  earth  we  can  always 
reduce  it  to  the  sun,  because  we  know  the  proportion  between  the 
distances  of  the  planet  from  the  earth  and  from  the  sun. 

All  the  large  planets  outside  the  earth  have  satellites  ;  we  can 
therefore  determine  their  masses  in  this  simple  way.  The  earth 
having  also  a  satellite,  its  mass  could  be  determined  in  the  same 
way  but  for  the  circumstance  already  mentioned  that  we  cannot 
determine  the  distance  of  the  moon  in  planetary  units,  as  we  can 
the  distance  of  the  satellites  of  the  other  planets  from  their  pri- 
maries. 

The  planets  Mercury  and  Venus  have  no  satellites.  It  is  therefore 
necessary  to  determine  their  masses  by  their  influence  in  altering 
the  elliptic  motions  of  the  other  planets  round  the  sun.  The  altera- 
tions thus  produced  are  for  the  most  part  so  small  that  their  deter- 
mination is  a  practical  problem  of  some  difficulty.  Thus  the  action 
of  Mercury  on  the  neighboring  planet  Venus  rarely  changes  the  po- 
sition of  the  latter  by  more  than  one  or  two  seconds  of  arc,  unless 
we  compare  observations  more  than  a  century  apart.  But  regular 
and  accurate  observations  of  Venus  were  rarely  made  until  after  the 
beginning  of  this  century.  The  mass  of  Venus  is  best  determined 
by  the  influence  of  the  planet  in  changing  the  position  of  the  plane 
of  the  earth's  orbit.  Altogether,  the  determination  of  the  masses 
of  Mercury  and  Venus  presents  one  of  the  most  complicated  prob- 
lems with  which  the  mathematical  astronomer  has  to  deal. 


CHAPTER    X. 

THE  REFRACTION  AND  ABERRATION  OF  LIGHT. 

§    1.    ATMOSPHERIC   REFRACTION. 

WHEN  we  refer  to  the  place  of  a  planet  or  star,  we 
usually  mean  its  true  place — i.e.,  its  direction  from 
an  observer  situated  at  the  centre  of  the  earth,  consid- 
ered as  a  geometrical  point.  We  have  shown  in  the  sec- 
tion on  parallax  how  observations  which  are  necessarily 
taken  at  the  surface  of  the  earth  are  reduced  to  what  they 
would  have  been  if  the  observer  were  situated  at  the 
earth's  centre.  In  this,  however,  we  have  supposed  the 
star  to  appear  to  be  projected  on  the  celestial  sphere  in 
the  prolongation  of  the  line  joining  the  observer  and  the 
star.  The  ray  from  the  star  is  considered  as  if  it  suffered 
no  deflection  in  passing  through  the  stellar  spaces  and 
through  the  earth's  atmosphere.  But  from  the  principles 
of  physics,  we  know  that  such  a  luminous  ray  passing  from 
an  empty  space  (as  the  stellar  spaces  are),  and  through  an 
atmosphere,  must  suffer  a  refraction,  .as  every  ray  of  light 
is  known  to  do  in  passing  from  a  rare  into  a  denser 
medium.  As  we  see  the  star  in  the  direction  which  its 
light  beam  has  when  it  enters  the  eye — that  is,  as  we  pro- 
ject the  star  on  the  celestial  sphere  by  prolonging  this 
light  beam  backward  into  space — there  must  be  an  appar- 
ent displacement  of  the  star  from  refraction,  and  it  is 
this  which  we  are  to  consider. 

We  may  recall  a  few  definitions  from  physics.  The 
ray  which  leaves  the  star  and  impinges  on  the  outer  sur- 


REFRACTION.  235 

face  of  the  earth's  atmosphere  is  called  the  incident  ray  • 
after  its  deflection  by  the  atmosphere  it  is  called  the  re- 
fracted ray.  The  difference  between  these  directions  is 
called  the  astronomical  refraction.  If  a  normal  is  drawn 
(perpendicular)  to  the  surface  of  the  refracting  medium  at 
the  point  where  the  incident  raj  meets  it,  the  acute  angle 
between  the  incident  ray  and  the  normal  is  called  the 
angle  of  incidence,  and  the  acute  angle  between  the  nor- 
mal and  the  refracted  ray  is  called  the  angle  of  refraction. 
The  refraction  itself  is  the  difference  of  these  angles. 
The  normal  and  both  incident  and  refracted  rays  are  in 
the  same  vertical  plane.  In 
Fig.  69  SA  is  the  ray  incident 
upon  the  surface  B  A  of  the  re- 
fracting medium  E'  B  AN, 
A  C  is  the  refracted  ray,  M N 
the  normal,  SAM  and  CAN 
the  angles  of  incidence  and  re- 
fraction respectively.  Produce 
C  A  backward  in  the  direction 
A  Sf  :  8  A  S'  is  the  refraction. 

An  observer  at   C  will  see   the       piQ   71._REFBACTION. 
star  xS  as  it  it  were  at  S .     A  S 

is  the  apparent  direction  of  the  ray  from  the  star  $,  and 
S'  is  the  apparent  place  of  the  star  as  affected  by  refrac- 
tion. 

This  supposes  the  space  above  B  B'  in  the  figure  to  be 
entirely  empty  spaces,  and  the  earth's  atmosphere,  equally 
dense  throughout,  to  fill  the  space  below  B  B '.  In  fact,  how- 
ever, the  earth's  atmosphere  is  most  dense  at  the  surface  of 
the  earth,  and  gradually  diminishes  in  density  to  its  exterior 
boundary.  Therefore,  if  we  wish  to  represent  the  facts  as 
they  are,  we  must  suppose  the  atmosphere  to  be  divided 
into  a  great  number  of  parallel  layers  of  air,  and  by  as- 
suming an  infinite  number  of  these  we  may  also  assume  that 
throughout  each  of  them  the  air  is  equally  dense.  Hence 
the  preceding  figure  will  only  represent  the  refraction  at 


236  ASTRONOMY. 

a  single  one  of  these  layers..  It  follows  from  this  that  the 
path  of  a  ray  of  light  through  the  atmosphere  is  not  a 
straight  line  like  A  C,  but  a  curve.  We  may  suppose 
this  curve  to  be  represented  in  Fig.  70,  where  the  num- 
ber of  layers  has  been  taken  very  small  to  avoid  conf  usin^ 
the  drawing. 

Let  C  be  the  centre  and  A  a  point  of  the  surface  of  the 
earth  ;  let  S  be  a  star,  and  S  e  a  ray  from  the  star 
which  is  refracted  at  the  various  layers  into  which  we  sup- 
pose the  atmosphere  to  be  divided,  and  which  finally 


FIG.    72. — REFRACTION   OF  LAYERS  OF  AIR. 

enters  the  eye  of  an  observer  at  A  in  the  apparent  direc- 
tion A  S'.  He  will  then  see  the  star  in  the  direction  S' 
instead  of  that  of  S-S,  and  S  A  £',  the  refraction,  will 
throw  the  star  nearer  to  the  zenith  Z. 

The  angle  8'  A  Z  is  the  apparent  zenith  distance  of  S ; 
the  true  zenith  distance  of  S  is  Z  A  S,  and  this  may  be 
assumed  to  coincide  with  8 'e,  as  for  all  heavenly  bodies 
except  the  moon  it  practically  does.  The  line  Se  pro- 
longed will  meet  the  line  A  Z  in  a  point  above  A,  sup- 
pose at  I '. 


REFRACTION. 


237 


Law  of  Refraction.  —  A  consideration  of  the  physical  condi- 
tions involved  has  led  to  the  following  form  for  the  refraction  in 
zenith  distance  (A  £), 


•m  which  C'  is  the  apparent  zenith  distance  of  the  star,  and  A  is  a 
constant  to  be  determined  by  observation.  A  is  found  to  be  about 
57",  so  that  we  may  write  (A  £)  =  57"  tan  C'  approximately. 

This  expression  gives  what  is  called  the  mean  refraction  —that  is, 
the  refraction  corresponding  to  a  mean  state  of  the  barometer  and 
thermometer.  It  is  clear  that  changes  in  the  temperature  and  pres- 
sure will  affect  the  density  of  the  air,  and  hence  its  refractive  power. 
The  tables  of  the  mean  refraction  made  by  BESSEL,  based  on  a  more 
accurate  formula  than  the  one  above,  are  now  usually  used,  and  these 
are  accompanied  by  auxiliary  tables  giving  the  small  corrections  for 
the  state  of  the  meteorological  instruments. 

Let  us  consider  some  of  the  consequences  of  refraction,  and  for 
our  purpose  we  may  take  the  formula  (A  Q  =  57  tan  C',  as  it 
very  nearly  represents  the  facts.  At  C'  =  0  (A  £)  =  0,  or  at  the 
apparent  zenith  there  is  no  refraction.  This  we  should  have  antici- 
pated as  the  incident  ray  in  itself  normal  to  the  refracting  surface. 

The  following  extract  from  a  refraction  table  gives  the  amount  of 
refraction  at  various  zenith  distances  : 


C' 

(AC) 

C' 

(AC) 

0° 

0'    0" 

70° 

2'    39" 

10° 

0'   10" 

80° 

5'    20" 

20° 

0'   33" 

85° 

10'     0" 

45° 

0'   58" 

88° 

18'     0" 

50° 

r  09" 

89° 

24'    25" 

60° 

1'   40" 

90° 

34'    30" 

Quantity  and  Effects  of  Refraction. — At  45°  the  refrac- 
tion is  about  I',  and  at  90°  it  is  34'  30" — that  is,  bodies  at 
the  zenith  distances  of  45°  and  90°  appear  elevated  above 
their  true  places  by  V  and  34J'  respectively.  If  the  sun 
has  just  risen — that  is,  if  its  lower  limb  is  just  in  apparent 
contact  with  the  horizon,  it  is,  in  fact,  entirely  below  the 
true  horizon,  for  the  refraction  (35')  has  elevated  its  cen- 
tre by  more  than  its  whole  apparent  diameter  (32'). 

The  moon  is  full  when  it  is  exactly  opposite  the  sun, 
and  therefore  were  there  no  atmosphere,  moon-rise  of  a 
full  moon  and  sunset  would  be  simultaneous.  In  fact, 


238  ASTRONOMY. 

both  bodies  being  elevated  by  refraction,  we  see  the  full 
moon  risen  before  the  sun  has  set.  On  April  20th,  1837, 
the  full  moon  rose  eclipsed  before  the  sun  had  set. 

We  see  from  the  table  that  the  refraction  varies  com- 
paratively little  between  0°  and  60°  of  zenith  distance,  but 
that  beyond  80°  or  85°  its  variation  is  quite  rapid. 

The  refraction  on  the  two  limbs  of  the  sun  or  moon  will 
then  be  different,  and  of  course  greater  on  the  lower  limb. 
This  will  apparently  be  lifted  up  toward  the  upper  limb 
more  than  the  upper  limb  is  lifted  away  from  it,  and 
hence  the  sun  and  moon  appear  oval  in  shape  when  near 
the  horizon.  For  example,  if  the  zenith  distance  of  the 
sun's  lower  limb  is  85°,  that  of  the  upper  will  be  about 
84°  28',  and  the  refractions  from  the  tables  for  these  two 
zenith  distances  differ  by  V  ;  therefore,  the  sun  will  ap- 
pear oval  in  sha'pe,  with  axes  of  32'  and  31'  approxi- 
mately. 

Determination  of  Refraction. — If  we  know  the  law  according 
to  which  refraction  varies — that  is,  if  we  have  an  accurate  formula 
which  will  give  ( A  Q  in  terms  of  £  we  can  determine  the  absolute 
refraction  for  any  one  point,  and  from  the  law  deduce  it  for  any 
other  points.  Thus  knowing  the  horizontal  refraction,  or  the  re- 
fraction in  the  horizon,  we  can  determine  the  refraction  at  other 
known  zenith  distances. 

We  know  the  time  of  (theoretical  or  true)  sunrise  and  sunset  by 
the  formula  of  §  7,  p.  44,  and  we  may  observe  the  time  of  apparent 
rising  and  setting  of  the  sun  (or  a  star).  The  difference  of  these 
times  gives  a  means  of  determining  the  effect  of  refraction. 

Or,  in  the  observations  for  latitude  by  the  method  of  §  8,  p.  47,  we 
can  measure  the  apparent  polar  distances  of  a  circumpolar  star  at 
its  upper  and  lower  culmination.  Its  polar  distances  above  and 
below  pole  should  be  equal ;  if  there  were  no  refraction  they  would 
be  so,  but  they  really  differ  by  a  quantity  which  it  is  easy  to  see  is 
the  difference  of  the  refractions  at  lower  and  upper  culminations. 
By  choosing  suitable  circumpolar  stars  at  various  polar  distances, 
this  difference  may  be  determined  for  all  polar  distances,  and  there- 
fore at  all  zenith  distances. 

§  2.  ABERRATION  AND  THE  MOTION  OP  LIGHT. 

Besides  refraction,  there  is  another  cause  which  prevents 
our  seeing  the  celestial  bodies  exactly  in  the  true  direction 
in  which  they  lie  from  us — namely,  the  progressive  mo- 


ABERRATION.  239 

tion  of  light.  We  now  know  that  we  see  objects  only 
by  the  light  which  emanates  from  them  and  reaches  our 
eyes,  and  we  also  know  that  this  light  requires  time  to 
pass  over  the  space  which  separates  us  from  the  object. 
After  the  ray  of  light  once  leaves  the  object,  the  latter 
may  move  away,  or  even  be  blotted  out  of  existence,  but 
the  ray  of  light  will  continue  on  its  course.  Consequent- 
ly when  we  look  at  a  star,  we  do  not  see  the  star  that  now 
is,  but  the  star  that  was  several  years  ago.  If  it  should  be 
annihilated,  we  should  still  see  it  during  the  years  which 
would  be  required  for  the  last  ray  of  light  emitted  by  it  to 
reach  us.  The  velocity  of  light  is  so  great  that  in  all  ob- 
servations of  terrestrial  objects,  our  vision  may  be  regarded 
as  instantaneous.  But  in  celestial  observations  the  time 
required  for  the  light  to  reach  us  is  quite  appreciable  and 
measurable. 

The  discovery  of  the  propagation  of  light  is  among  the 
most  remarkable  of  those  made  by  modern  science.  The 
fact  that  light  requires  time  to  travel  was  first  learned  by 
the  observations  of  the  satellites  of  Jupiter.  Owing  to 
the  great  magnitude  of  this  planet,  it  casts  a  much  longer 
and  larger  shadow  than  our  earth  does,  and  its  inner  sat- 
ellite is  therefore  eclipsed  at  every  revolution.  These 
eclipses  can  be  observed  from  the  earth,  the  satellite  van- 
ishing from  view  as  it  enters  the  shadow,  and  suddenly 
reappearing  when  it  leaves  it  again.  The  accuracy  with 
which  the  times  of  this  disappearance  and  reappearance 
could  be  observed,  and  the  consequent  value  of  such  ob- 
servations for  the  determination  of  longitudes,  led  the 
astronomers  of  the  seventeenth  century  to  make  a  careful 
study  of  the  motions  of  these  bodies.  It  was,  however, 
necessary  to  make  tables  by  which  the  times  of  the  eclipses 
could  be  predicted.  It  was  found  by  ROEMER  that  these 
times  depended  on  the  distance  of  Jupiter  from  the  earth. 
If  he  made  his  tables  agree  with  observations  when  the 
earth  was  nearest  Jupiter,  it  was  found  that  as  the  earth 
receded  from  Jupiter  in  its  annual  course  around  the  sun, 


240  ASTRONOMY. 

the  eclipses  were  constantly  seen  later,  until,  when  at  its 
greatest  distance,  the  times  appeared  to  be  22  minutes  late. 
ROEMER  saw  that  it  was  in  the  highest  degree  improbable 
that  the  actual  motions  of  the  satellites  should  be  affected 
with  any  such  inequality  ;  he  therefore  propounded  the 
bold  theory  that  it  took  time  for  light  to  come  from  Ju- 
piter to  the  earth.  The  extreme  differences  in  the  times 
of  the  eclipse  being  22  minutes,  he  assigned  this  as  the  time 
required  for  light  to  cross  the  orbit  of  the  earth,  and  so 
concluded  that  it  came  from  the  sun  to  the  earth  in  1 1 
minutes.  We  now  know  that  this  estimate  was  too  great, 
and  that  the  true  time  for  this  passage  is  about  8  minutes 
and  18  seconds. 

Discovery  of  Aberration. — At  first  this  theory  of  ROK- 
MER  was  not  fully  accepted  by  his  contemporaries.  But 
in  the  year  1729  the  celebrated  BRADLEY,  afterward  As- 
tronomer Royal  of  England,  discovered  a  phenomenon  of 
an  entirely  different  character,  which  confirmed  the  theory. 
He  was  then  engaged  in  making  observations  on  the  star 
y  Dravonis  in  order  to  determine  its  parallax.  The  effect 
of  parallax  would  have  been  to  make  the  declination 
greatest  in  June  and  least  in  December,  while  in  March 
and  September  the  star  would  occupy  an  intermediate  or 
mean  position.  But  the  result  was  entirely  different. 
The  declinations  of  June  and  December  were  the  same, 
showing  no  effect  of  parallax  ;  but  instead  of  remaining 
constant  the  rest  of  the  year,  the  declination  was  some  40 
seconds  greater  in  September  than  in  March,  when  the 
effect  of  parallax  would  be  the  same.  This  showed  that 
the  direction  of  the  star  appeared  different,  not  according 
to  the  position  of  the  earth,  but  according  to  the  direction 
of  its  motion  around  the  sun,  the  star  being  apparently 
displaced  in  this  direction. 

It  has  been  said  that  the  explanation  of  this  singular 
anomaly  was  first  suggested  to  BRADLEY  while  sailing  on 
the  Thames.  He  noticed  that  when  his  boat  moved  rapid- 
ly at  right  angles  to  the  true  direction  of  the  wind,  the 


ABERRATION.  241 

apparent  direction  of  the  wind  changed  toward  the  point 
whither  the  boat  was  going.  When  the  boat  sailed  in  an 
opposite  direction,  the  apparent  direction  of  the  wind  sud- 
denly changed  in  a  corresponding  way.  Here  was  a  phe- 
nomenon very  analogous  to  that  which  he  had  observed  in 
the  stars,  the  direction  from  which  the  wind  appeared  to 
come  corresponding  to  the  direction  in  which  the  light 
reached  the  eye.  This  direction  changed  with  the  mo- 
tion of  the  observer  according  to  the  same  law  in  the  two 
cases.  He  now  saw  that  the  apparent  displacement  of  the 
star  was  due  to  the  motion  of  the  rays  of  light  combined 
with  that  of  the  earth  in  its  orbit,  the  apparent  direction 
of  the  star  depending,  not  upon  the  absolute  direction 
from  which  the  ray  comes,  but  upon  the  relation  of  this 
direction  to  the  motion  of  the  observer. 

To  show  how  this  is,  let  A  B  be  the  optical  axis  of  a 
telescope,  and  /S  a  star  from  which  emanates  a  ray  mov- 
ing in  the  true  direction  S  A  Bf. 
Perhaps  the  reader  will  have  a  clearer 
conception  of  the  subject  if  he  imag- 
ines A  B  to  be  a  rod  which  an  ob- 
server at  13  seeks  to  point  at  the  star 
S.  It  is  evident  that  he  will  point 
this  rod  in  such  a  way  that  the  ray 
of  light  shall  run  accurately  along  its 
length.  Suppose  now  that  the  ob- 
server is  moving  from  B  toward  B' 
with  such  a  velocity  that  he  moves 
from  B  to  B'  during  the  time  re-  _ 

quired  for  a  ray  of  light  to  move  from 
A  to  B'.  Suppose  also  that  the  ray  of  light  S  A  reaches 
A  at  the  same  time  that  the  end  of  his  rod  does.  Then 
it  is  clear  that  while  the  rod  is  moving  from  the  position 
A  B  to  the  position  A  B ',  the  ray  of  light  will  move  from 
A  to  B',  and  will  therefore  run  accurately  along  the  length 
of  the  rod.  For  instance,  if  b  is  one  third  of  the  way 
from  B  to  B',  then  the  light,  at  the  instant  of  the  rod  tak- 


242  ASTRONOMY. 

ing  the  position  b  a,  will  be  one  third  of  the  way  from  A 
to  B ',  and  will  therefore  be  accurately  on  the  rod.  Con- 
sequently, to  the  observer,  the  rod  will  appear  to  be  point- 
ed at  the  star.  In  reality,  however,  the  pointing  will  not 
be  in  the  true  direction  of  the  star,  but  will  devaate  from 
it  by  an  angle  of  which  the  tangent  is  the  ratio  of  the 
velocity  with  which  the  observer  is  carried  along  to  the 
velocity  of  light.  This  presupposes  that  the  motion  of  the 
observer  is  at  right  angles  to  that  of  a  ray  of  light.  If 
this  is  not  his  direction,  we  must  resolve  his  velocity  into 
two  components,  one  at  right  angles  to  the  ray  and  one 
parallel  to  it.  The  latter  will  not  affect  the  apparent  di- 
rection of  the  star,  which  will  therefore  depend  entirely 
upon  the  former. 

Effects  of  Aberration. — The  apparent  displacement  of 
the  heavenly  bodies  thus  produced  is  called  the  aberration 
of  light.  Its  effect  is  to  cause  each  of  the  fixed  stars  to 
ascribe  an  apparent  annual  oscillation  in  a  very  small  or- 
bit. The  nature  of  the  displacement  may  be  conceited 
of  in  the  following  way  :  Suppose  the.  earth  at  any  moment, 
in  the  course  of  its  annual  revolution,  to  be  moving  to- 
ward a  point  of  the  celestial  sphere,  which  we  may  call  P. 
Then  a  star  lying  in  the  direction  P  or  in  the  opposite  di- 
rection will  suffer  no  displacement  whatever.  A  star  ly- 
ing in  any  other  direction  will  be  displaced  in  the  direc- 
tion of  the  point  P  by  an  angle  proportional  to  the  sine  of 
its  angular  distance  from  P.  At  90°  from  P  the  dis- 
placement will  be  a  maximum,  and  its  angular  amount 
will  be  such  that  its  tangent  will  be  equal  to  the  ratio  of 
the  velocity  of  the  earth  to  that  of  light.  If  A  be  the 
"  aberration"  of  the  star,  and  P  S  its  angular  distance 
from  the  point  P,  we  shall  have, 

tan  A  =  - -,  sin  PS, 

v 

v'  and  v  being  the  respective  velocities  of  light  and  of  the 
earth. 


VELOCITY  OF  LIGHT.  243 

Now,  if  the  star  lies  near  the  pole  of  the  ecliptic,  its  di- 
rection will  always  be  nearly  at  right  angles  to  the  direc- 
tion in  which  the  earth  is  moving.  A  little  consideration 
will  show  that  it  will  seem  to  describe  a  circle  in  conse- 
quence of  aberration.  If,  however,  it  lies  in  the  plane  of 
the  earth's  orbit,  then  the  various  points  toward  which 
the  earth  moves  in  the  course  of  the  year  all  lying  in  the 
ecliptic,  and  the  star  being  in  this  same  plane,  the  appar- 
ent motion  will  be  an  oscillation  back  and  forth  in  this 
plane,  and  in  all  other  positions  the  apparent  motion  will 
be  in  an  ellipse  more  and  more  flattened  as  we  approach 
the  ecliptic. 

Velocity  of  Light. — The  amount  of  aberration  can  be 
determined  in  two  ways.  If  we  know  the  time  which 
light  requires  to  come  from  the  sun  to  the  earth,  a  simple 
calculation  will  enable  us  to  determine  the  ratio  between 
this  velocity  and  that  of  the  earth  in  its  orbit.  For  in- 
stance, suppose  the  time  to  be  498  seconds  ;  then  light 
will  cross  the  orbit  of  the  earth  in  996  seconds.  The  cir- 
cumference of  the  earth  being  found  by  multiplying  its 
diameter  by  3-1416,  we  thus  find  that,  on  the  supposition 
we  have  made,  light  would  move  around  the  circumfer- 
ence of  the  earth's  orbit  in  52  minutes  and  8  seconds. 
But  the  earth  makes  this  same  circuit  in  365J  days,  and 
the  ratio  of  these  two  quantities  is  10090.  The  maximum 
displacement  of  the  star  by  aberration  will  therefore  be  the 
angle  of  which  the  tangent  is  3-0^-^,  and  this  angle  we 
find  by  trigonometrical  calculation  to  be  20" -44. 

This  calculation  presupposes  that  we  know  how  long 
light  requires  to  come  from  the  sun.  This  is  not  known 
with  great  accuracy  owing  to  the  unavoidable  errors  with 
which  the  observations  of  Jupiter**  satellites  are  affected. 
It  is  therefore  more  usual  to  reverse  the  process  and  de- 
termine the  displacement  of  the  stars  by  direct  observa- 
tion, and  then,  by  a  calculation  the  reverse  of  that  we 
have  just  made,  to  determine  the  time  required  by  light 
to  reach  us  from  the  sun.  Many  painstaking  determina- 


244  ASTRONOMY. 

tions  of  this  quantity  have  been  made  since  the  time  of 
BRADLEY,  and  as  the  result  of  them  we  may  say  that  the 
value  of  the  "  constant  of  aberration,"  as  it  is  called,  is 
certainly  between  20"  •  4  and  20"  •  5  ;  the  chances  are  that  it 
does  not  deviate  from  20".  44  by  more  than  two  or  three 
hundredths  of  a  second. 

It  will  be  noticed  that  by  determining  the  constant  of 
aberration,  or  by  observing  the  eclipses  of  the  satellites  of 
Jupiter,  we  may  infer  the  time  required  for  light  to  pass 
from  the  sun  to  the  earth.  But  we  cannot  thus  determine 
the  velocity  of  light  unless  we  know  how  far  the  sun  is. 
The  connection  between  this  velocity  and  the  distance  of 
the  sun  is  such  that  knowing  one  we  can  infer  the  other. 
Let  us  assume,  for  instance,  that  the  time  required  for 
light  to  reach  us  from  the  sun  is  498  seconds,  a  time  which 
is  probably  accurate  within  a  single  second.  Then  know- 
ing the  distance  of  the  sun,  we  may  obtain  the  velocity  of 
light  by  dividing  it  by  498.  But,  on  the  other  hand,  if  we 
can  determine  how  many  miles  light  moves  in  a  second,  we 
can  thence  infer  the  distance  of  the  sun  by  multiplying  it 
by  the  same  factor.  During  the  last  century  the  distance 
of  the  sun  was  found  to  be  certainly  between  90  and  100 
millions  of  miles.  It  was  therefore  correctly  concluded 
that  the  velocity  of  light  was  something  less  than  200,000 
miles  per  second,  and  probably  between  180,000  and 
200,000.  This  velocity  has  since  been  determined  more 
exactly  by  the  direct  measurements  at  the  surface  of  the 
earth  already  mentioned. 


CHAPTER  XI. 

CHRONOLOGY. 
§   1.    ASTRONOMICAL  MEASURERS    OF  TIME. 

THE  most  intimate  relation  of  astronomy  to  the  daily 
life  of  mankind  has  always  arisen  from  its  affording  the 
only  reliable  and  accurate  measure  of  long  intervals  of  time. 
The  fundamental  units  of  time  in  all  ages  have  been  the 
day,  the  month,  and  the  year,  the  first  being  measured  by 
the  revolution  of  the  earth  on  its  axis,  the  second,  prim- 
itively, by  that  of  the  rnoon  around  the  earth,  and  the  third 
by  that  of  the  earth  round  the  sun.  Had  the  natural  month 
consisted  of  an  exact  entire  number  of  days,  and  the  year 
of  an  exact  entire  number  of  months,  there  would  have 
been  no  history  of  the  calendar  to  write.  There  being  no 
such  exact  relations,  innumerable  devices  have  been  tried 
for  smoothing  off  the  difficulties  thus  arising,  the  mere 
description  of  which  would  fill  a  volume.  We  shall  en- 
deavor to  give  the  reader  an  idea  of  the  general  character 
of  these  devices,  including  those  from  which  our  own  cal- 
endar originated,  without  wearying  him  by  the  introduc- 
tion of  tedious  details. 

Of  the  three  units  of  time  just  mentioned^  the  most  nat- 
ural and  striking  is  the  shortest — namely,  the  day.  Mark- 
ing as  it  does  the  regular  alternations  of  wakefulness  and 
rest  for  both  maa  and  animals,  no  astronomical  observa- 
tions were  necessary  to  its  recognition.  It  is  so  nearly 
uniform  in  length  that  the  most  refined  astronomical  obser- 
vations of  modern  times  have  never  certainly  indicated 


246  ASTRONOMY. 

any  change.  This  uniformity,  and  its  entire  freedom  from 
all  ambiguity  of  meaning,  have  always  made  the  day  a 
common  fundamental  unit  of  astronomers.  Except  for 
the  inconvenience  of  keeping  count  of  the  great  number 
of  days  between  remote  epochs,  no  greater  unit  would 
ever  have  been  necessary,  and  we  might  all  date  our  let- 
ters by  the  number  of  days  after  CHRIST,  or  after  a  sup- 
posed epoch  of  creation. 

The  difficulty  of  remembering  great  numbers  is  such 
that  a  longer  unit  is  absolutely  necessary,  even  in  keeping 
the  reckoning  of  time  for  a  single  generation.  Such  a 
unit  is  the  year.  The  regular  changes  of  seasons  in  all  ex- 
tra-tropical latitudes  renders  this  unit  second  only  to  the 
day  in  the  prominence  with  which  it  must  have  struck  the 
minds  of  primitive  man.  These  changes  are,  however,  so 
slow  and  ill-marked  in  their  progress,  that  it  would  have 
been  scarcely  possible  to  make  an  accurate  determination 
of  the  length  of  the  year  from  the  observation  of  the  sea- 
sons. Here  astronomical  observations  came  to  the  aid  of 
our  progenitors,  and,  before  the  beginning  of  extant  his- 
tory, it  was  known  that  the  alternation  of  seasons  was  due 
to  the  varying  declination  of  the  sun,  as  the  latter  seemed 
to  perform  its  annual  course  among  the  stars  in  the 
"  oblique  circle"  or  ecliptic.  The  common  people,  who  did 
not  understand  the  theory  of  the  sun's  motion,  knew  that 
certain  seasons  were  marked  by  the  position  of  certain 
bright  stars  relatively  to  the  sun — that  is,  by  those  stars 
rising  or  setting  in  the  morning  or  evening  twilight. 
Thus  arose  two  methods  of  measuring  the  length  of  the 
year — the  one  by  the  time  when  the  sun  crossed  the  equi- 
noxes or  solstices,  the  other  when  it  seemed  to  pass  a  cer- 
tain point  among  the  stars.  As  we  have  already  explain- 
ed, these  years  were  slightly  different,  owing  to  the  pre 
cession  of  the  equinoxes,  the  first  or  equinoctial  year  being 
a  little  less  and  the  second  or  sidereal  year  a  little  greater 
than  3651  days. 

The  number  of  days  in  a  year  is  too  great  to  admit  of 


CHRONOLOGY.  247 

their  being  easily  remembered  without  any  break  ;  an  in- 
termediate period  is  therefore  necessary.  Such  a  period 
is  measured  by  the  revolution  of  the  moon  around  the 
earth,  or,  more  exactly,  by  the  recurrence  of  new  moon, 
which  takes  place,  on  the  average,  at  the  end  of  nearly 
29£  days.  The  nearest  round  number  to  this  is  30  days, 
and  12  periods  of  30  days  each  only  lack  5J  days  of  being 
a  year.  It  has  therefore  been  common  to  consider  a  year 
as  made  up  of  12  months,  the  lack  of  exact  correspondence 
being  filled  by  various  alterations  of  the  length  of  the 
month  or  of  the  year,  or  by  adding  surplus  days  to  each 
year. 

The  true  lengths  of  the  day,  the  month,  and  the  year 
having  no  common  divisor,  a  difficulty  arises  in  attempting 
to  make  months  or  days  into  years,  or  days  into  months, 
owing  to  the  fractions  which  will  always  be  left  over.  At 
the  same  time,  some  rule  bearing  on  the  subject  is  necessary 
in  order  that  people  may  be  able  to  remember  the  year, 
month,  and  day.  Such  rules  are  found  by  choosing  some 
cycle  or  period  which  is  very  nearly  an  exact  number  of 
two  units,  of  months  and  of  days  for  example,  and  by  di- 
viding this  cycle  up  as  evenly  as  possible.  The  principle 
on  which  this  is  done  can  be  seen  at  once  by  an  example, 
for  which  we  shall  choose  the  lunar  month.  The  true 
length  of  this  month  is  29  •  5305884  days.  We  see  that 
two  of  these  months  is  only  a  little  over  59  days  ;  so,  if 
we  take  a  cycle  of  59  days,  and  divide  it  into  two  months, 
the  one  of  80  and  the  other  of  29  days,  we  shall  have  a 
first  approximation  to  a  true  average  month.  But  our 
cycle  will  be  too  short  by  Od  •  061,  the  excess  of  two  months 
over  59  days,  and  this  error  will  be  added  at  the  end  of 
every  cycle,  and  thus  go  on  increasing  as  long  as  the  cycle 
is  used  without  change.  At  the  end  of  16  cycles,  or  of 
32  lunar  months,  the  accumulated  error  will  amount  to 
one  day.  At  the  end  of  this  time,  if  not  sooner,  we 
should  have  to  add  a  day  to  one  of  the  months. 

Seeing  that  we  shall  ultimately  be  wrong  if  we  have  a 


ASTRONOMY. 

two-month  cycle,  we  seek  for  a  more  exact  one.  Each 
month  of  30  days  is  nearly  0d  •  4T  too  long,  and  each  month 
of  29  days  is  rather  more  than  Od  -  53  too  short.  So  in  the 
long  run  the  months  of  30  days  ought  to  be  more  numer- 
ous than  those  of  29  days  in  the  ratio  that  53  bears  to 
4T,  or,  more  exactly,  in  the  ratio  that  •  5305884  bears  to 
•4694116.  A  close  approximation  will  be  had  by  having 
the  long  months  one  eighth  more  numerous  than  the  short 
ones,  the  numbers  in  question  being  nearly  in  the  ratio  of 
9  :  8.  So,  if  we  take  a  cycle  of  IT  months,  9  long  and  8 
short  ones,  we  find  that  9  x  30  +  8  x  29  =  502  days  for 
the  assumed  length  of  our  cycle,  whereas  the  true  length 
of  IT  months  is  very  near  502d-0200.  The  error  will 
therefore  be  •  02  of  a  day  for  every  cycle,  and  will  not 
amount  to  a  day  till  the  end  of  50  cycles,  or  nearly  TO 
years. 

A  still  nearer  approach  will  be  found  by  taking  a  cycle 
of  49  months,  26  to  be  long  and  23  short  ones.  These 
49  months  will  be  composed  of  26  x  30  +  23  x  29  = 
144T  days,  whereas  49  true  lunar  months  will  comprise 
1446  •  998832  days.  Each  cycle  will  therefore  be  too  long 
by  only  -001168  of  a  day,  and  the  error  would  not  amount 
to  a  day  till  the  end  of  84  cycles,  or  more  than  3000  years. 

Although  these  cycles  are  so  near  the  truth,  they  could 
not  be  used  with  convenience  because  they  would  begin 
at  different  times  of  the  year.  The  problem  is  therefore 
to  find  a  cycle  which  shall  comprise  an  entire  number  of 
years.  We  shall  see  hereafter  what  solutions  of  this 
problem  were  actually  found. 

§  2.    FORMATION  OP  CALENDARS. 

The  months  now  or  heretofore  in  use  among  the  peoples 
of  the  globe  may  for  the  most  part  be  divided  into  two 
classes  : 

(1.)  The  lunar  month  pure  and  simple,  or  the  mean 
interval  between  successive  new  moons, 


THE  CALENDAR,  249 

(2.)  An  approximation  to  the  twelfth  part  of  a  year, 
without  respect  to  the  motion  of  the  moon. 

The  Lunar  Month. — The  mean  interval  between  con- 
secutive new  moons  being  nearly  29£  days,  it  was  common 
in  the  use  of  the  pure  lunar  month  to  have  months  of  29  and 
30  days  alternately.  This  supposed  period,  however,  as  just 
shown,  will  fall  short  by  a  day  in  about  2J  years.  This  de- 
fect was  remedied  by  introducing  cycles  containing  rather 
more  months  of  30  than  of  29  days,  the  small  excess  of 
long  months  being  spread  uniformly  through  the  cycle. 
Thus  the  Greeks  had  a  cycle  of  235  months  (to  be  soon 
described  more  fully),  of  which  125  were  full  or  long 
months,  and  110  were  short  or  deficient  ones.  We  see 
that  the  length  of  this  cycle  was  6940  days  (125  x  30  + 
110  x  29),  whereas  the  length  of  235  true  lunar  months 
is  235  x  29  -  53088  =  6939  -  688  days.  The  cycle  was  there- 
fore too  long  by  less  than  one  third  of  a  day,  and  the  error 
of  count  would  amount  to  only  one  day  in  more  than  TO 
years.  The  Mohammedans,  again,  took  a  cycle  of  360 
months,  which  they  divided  into  169  short  and  191  long 
ones.  The  length  of  this  cycle  was  10631  days,  while  the 
true  length  of  360  lunar  months  is  10631-012  days.  The 
count  would  therefore  not  be  a  day  in  error  until  the  end  of 
about  80  cycles,  or  nearly  23  centuries.  This  month  there- 
fore follows  the  moon  closely  enough  for  all  practical  pur- 
poses. 

Months  other  than  Lunar. — The  complications  of  the 
system  just  described,  and  the  consequent  difficulty  of 
making  the  calendar  month  represent  the  course  of  the 
moon,  are  so  great  that  the  pure  lunar  month  was  gen- 
erally abandoned,  except  among  people  whose  religion  re- 
quired important  ceremonies  at  the  time  of  new  moon. 
In  cases  of  such  abandonment,  the  year  has  been  usually 
divided  into  12  months  of  slightly  different  lengths.  The 
ancient  Egyptians,  however,  had  12  months  of  30  days 
each,  to  which  they  added  5  supplementary  days  at  the 
close  of  each  year. 


250  ASTRONOMY. 

Kinds  of  Year. — As  we  find  two  different  systems  of 
months  to  have  been  used,  so  we  may  divide  the  calendar 
years  into  three  classes — namely  : 

(1.)  The  lunar  year,  of  12  lunar  months. 

(2.)  The  solar  year. 

(3.)  The  combined  luni-solar  year. 

The  Lunar  Year. — We  have  already  called  attention  to 
the  fact  that  the  time  of  recurrence  of  the  year  is  not  well 
marked  except  by  astronomical  phenomena  which  the 
casual  observer  would  hardly  remark.  But  the  time  of 
new  moon,  or  of  beginning  of  the  month,  is  always  well 
marked.  Consequently,  it  was  very  natural  for  people  to 
begin  by  considering  the  year  as  made  up  of  twelve  luna- 
tions, the  error  of  eleven  days  being  unnoticeable  in  a 
single  year,  unless  careful  astronomical  observations  were 
made.  Even  when  this  error  was  fully  recognized ,  it  might 
be  considered  better  to  use  the  regular  year  of  12  lunar 
months  than  to  use  one  of  an  irregular  or  varying  number 
of  months.  Such  a  year  is  the  religious  one  of  the  Mo- 
hammedans to  this  day.  The  excess  of  11  days  will 
amount  to  a  whole  year  in  33  years,  32  solar  years  being 
nearly  equal  to  33  lunar  years.  In  this  period  therefore 
each  season  will  have  coursed  through  all  times  of  the 
year.  The  lunar  year  has  therefore  been  called  the 
' '  wandering  year. ' ' 

The  Solar  Year. — In  forming  this  year,  the  attempt  to 
measure  the  year  by  revolutions  of  the  moon  is  entirely 
abandoned,  and  its  length  is  made  to  depend  entirely  on 
the  change  of  the  seasons.  The  solar  year  thus  indicated 
is  that  most  used  in  both  ancient  and  modern  times.  Its 
length  has  been  known  to  be  nearly  365J  days  from  the 
times  of  the  earliest  astronomers,  and  the  system  adopted 
in  our  calendar  of  having  three  years  of  365  days  each,  fol- 
lowed by  one  of  366  days,  has  been  employed  in  China 
from  the  remotest  historic  times.  This  year  of  365^  days 
is  now  called  by  us  the  Julian  Year,  after  JULIUS  CAESAR, 
from  whom  we  obtained  it. 


THE  CALENDAR.  251 

The  Luni-Solar  Year. — If  the  lunar  months  must,  in 
some  way,  be  made  up  into  solar  years  of  the  proper  av- 
erage length,  then  these  years  must  be  of  unequal  length, 
some  having  twelve  months  and  others  thirteen.  Thus,  a 
period  or  cycle  of  eight  years  might  be  made  up  of  '99 
lunar  months,  5  of  the  years  having  12  months  each,  and 
3  of  them  13  months  each.  Such  a  period  would  comprise 
2923J  days,  so  that  the  average  length  of  the  year  would 
be  365  days  10£  hours.  This  is  too  great  by  about  4  hours 
42  minutes.  This  very  plan  was  proposed  in  ancient 
Greece,  but  it  was  superseded  by  the  discovery  of  the 
3£etonic  Cycle,  which  figures  in  our  church  calendar  to 
this  day.  A  luni-solar  year  of  this  general  character  was 
also  used  by  the  Jews. 

The  Metonic  Cycle. — The  preliminary  considerations  we 
have  set  forth  will  now  enable  us  to  understand  the  origin 
of  our  own  calendar.  We  begin  with  the  Metonic  Cycle 
of  the  ancient  Greeks,  which  still  regulates  some  religious 
festivals,  although  it  has  disappeared  from  our  civil  reck- 
oning of  time.  The  necessity  of  employing  lunar  months 
caused  the  Greeks  great  difficulty  in  regulating  their  cal- 
endar so  as  to  accord  with  their  rules  for  religious  feasts, 
until  a  solution  of  the  problem  was  found  by  METON,  about 
433  B.C.  The  great  discovery  of  METON  was  that  a  period 
or  cycle  of  6940  days  could  be  divided  up  into  235  lunar 
months,  and  also  into  19  solar  years.  Of  these  months, 
125  were  to  be  of  30  days  each,  and  110  of  29  days  each, 
which  would,  in  all,  make  up  the  required  6940  days.  To 
see  how  nearly  this  rule  represents  the  actual  motions  of 
the  sun  and  moon,  we  remark  that : 

Days.  Hours.        Min. 

235  lunations  require  6939         16         31 

19  Julian  years    "  6939         18  0 

19  true  solar  years  require     6939         14         27 

We  see  that  though  the  cycle  of  6940  days  is  a  few  hours 
too  long,  yet,  if  we  take  235  true  lunar  months,  we  find 


252  ASTRONOMY. 

their  whole  duration  to  be  a  little  less  than  19  Julian  years  of 
365J  days  each,  and  a  little  inore  than  19  true  solar  years. 

The  problem  now  was  to  take  these  235  months  and  divide 
them  up  into  19  years,  of  which  12  should  have  12  months 
each,  and  7  should  have  13  months  each.  The  long  years, 
or  those  of  13  months,  were  probably  those  corresponding 
to  the  numbers  3,  5,  8,  11,  13,  16,  and  19,  while  the  first, 
second,  fourth,  sixth,  etc. ,  were  short  years.  In  general, 
the  months  had  29  and  30  days  alternately,  but  it  was 
necessary  to  substitute  a  long  month  for  a  short  one  every 
two  or  three  years,  so  that  in  the  cycle  there  should  be 
125  long  and  110  short  months. 

Golden  INTumber. — This  is  simply  the  number  of  the 
year  in  the  Metonic  Cycle,  and  is  said  to  owe  its  appella- 
tion to  the  enthusiasm  of  the  Greeks  over  METON'S  dis- 
covery, the  authorities  having  ordered  the  division  and 
numbering  of  the  years  in  the  new  calendar  to  be  in- 
scribed on  public  monuments  in  letters  of  gold.  The  rule 
for  finding  the  golden  number  is  to  divide  the  number  of 
the  year  by  19,  and  add  1  to  the  remainder.  From  1881 
to  1899  it  may  be  found  by  simply  subtracting  1880  from 
the  year.  It  is  employed  in  our  church  calendar  for  find- 
ing the  time  of  Easter  Sunday. 

Period  of  Gaily  pus. — We  have  seen  that  the  cycle  of 
6940  days  is  a  few  hours  too  long  either  for  235  lunar 
months  or  for  19  solar  years.  CALLYPUS  therefore  sought 
to  improve  it  by  taking  one  day  off  of  every  fourth  cycle, 
so  that  the  four  cycles  should  have  27759  days,  which 
were  to  be  divided  into  940  months  and  into  76  years. 
These  years  would  then  be  Julian  years,  while  the  recur- 
rence of  new  moon  would  only  be  six  hours  in  error  at  the 
end  of  the  76  years.  Had  he  taken  a  day  from  every 
third  cycle,  and  from  some  year  and  month  of  that  cycle, 
he  would  have  been  yet  nearer  the  truth. 

The  Mohammedan  Calendar. — Among  the  most  remark- 
able calendars  which  have  remained  in  use  to  the  present 
time  is  that  of  the  Mohammedans.  The  year  is  composed 


THE  MOHAMMEDAN  CALENDAR.  253 

of  12  lunar  months,  and  therefore,  as  already  mentioned, 
does  not  correspond  to  the  course  of  the  seasons.  As  with 
other  systems,  the  problem  is  to  find  such  a  cycle  that  an 
entire  number  of  these  lunar  years  shall  correspond  to  an 
integral  number  of  days.  Multiplying  the  length  of  the 
lunar  month  by  12,  we  find  the  true  length  of  the  lunar 
year  to  be  354  •  36T06  days.  The  fraction  of  a  day  being 
not  far  from  one  third,  a  three-year  cycle,  comprising  two 
years  of  354  and  one  of  355  days,  would  be  a  first  approx- 
imation to  three  lunar  years,  but  would  still  be  one  tenth 
of  a  day  too  short.  In  ten  such  cycles  or  thirty  years, 
this  deficiency  would  amount  to  an  entire  day,  and  by  add- 
ing the  day  at  the  end  of  each  tenth  three-year  cycle, 
a  very  near  approach  to  the  true  motion  of  the  moon 
will  be  obtained.  This  thirty -year  cycle  will  consist  of 
10631  days,  while  the  true  length  of  360  lunar  months  is 
10631  •  0116  days.  The  error  will  not  amount  to  a  day  until 
the  end  of  87  cycles,  or  2610  years,  so  that  this  system  is 
accurate  enough  for  all  practical  purposes.  The  common 
Mohammedan  year  of  354  days. is  composed  of  months 
containing  alternately  30  and  29  days,  the  first  having 
30  and  the  last  29.  In  the  years  of  355  days  the  alter- 
nation is  the  same,  except  that  one  day  is  added  to  the  last 
month  of  the  year. 

The  old  custom  was  to  take  for  the  first  day  of  the 
month  that  following  the  evening  on  which  the  new  moon 
could  first  be  seen  in  the  west.  It  is  said  that  before  the 
exact  arrangement  of  the  Mohammedan  calendar  had  been 
completed,  the  rule  was  that  the  visibility  of  the  crescent 
moon  should  be  certified  by  the  testimony  of  two  wit- 
nesses. The  time  of  new  moon  given  in  our  modern 
almanacs  is  that  when  the  moon  passes  nearly  between  us 
and  the  sun,  and  is  therefore  entirely  invisible.  The  moon 
is  generally  one  or  two  days  old  before  it  can  be  seen  in  the 
evening,  and,  in  consequence,  the  lunar  month  of  the  Mo- 
hammedans and  of  others  commences  about  two  days  after 
the  actual  almanac  time  of  new  moon. 


254  ASTRONOMY. 

The  civil  calendar  now  in  use  throughout  Christendom 
had  its  origin  among  the  Romans,  and  its  foundation  was 
laid  by  JULIUS  CAESAR.  Before  his  time,  Rome  can  hardly  be 
said  to  have  had  a  chronological  system,  the  length  of  the 
year  not  being  prescribed  by  any  invariable  rule,  and  be- 
ing therefore  changed  from,  time  to  time  to  suit  the  caprice 
or  to  compass  the  ends  of  the  rulers.  Instances  of  this 
tampering  disposition  are  familiar  to  the  historical  student. 
It  is  said,  for  instance,  that  the  Gauls  having  to  pay  a 
certain  monthly  tribute  to  the  Romans,  one  of  the  govern- 
ors ordered  the  year  to  be  divided  into  14  months,  in 
order  that  the  pay  days  might  recur  more  rapidly.  To 
remedy  this,  CJSSAR  called  in  the  aid  of  SOSIGENES,  an  as- 
tronomer of  the  Alexandrian  school,  and  by  them  it  was 
arranged  that  the  year  should  consist  of  365  days,  with  the 
addition  of  one  day  to  every  fourth  year.  The  old  Roman 
months  were  afterward  adjusted  to  the  Julian  year  in 
such  a  way  as  to  give  rise  to  the  somewhat  irregular 
arrangement  of  months  which  we  now  have. 

Old  and  New  Styles.— The  mean  length  of  the  Julian 
year  is  365 J  days,  about  11J  minutes  greater  than  that  of 
the  true  equinoctial  year,  which  measures  the  recurrence 
of  the  seasons.  This  difference  is  of  little  practical  im- 
portance, as  it  only  amounts  to  a  week  in  a  thousand  years, 
and  a  change  of  this  amount  in  that  period  is  productive 
of  no  inconvenience.  But,  desirous  to  have  the  year  as 
correct  as  possible,  two  changes  were  introduced  into  the 
calendar  by  Pope  GREGORY  XIII.  with  this  object.  They 
were  as  follows  : 

1.  The  day  following  October  4,  1582,  was  called  the 
15th  instead  of  the  5th,  thus  advancing  the  count  10  days. 

2.  The  closing  year  of  each  century,  1600,  1700,  etc., 
instead  of  being  always  a  leap  year,   as    in  the  Julian 
calendar,  is  such  only  when  the  number  of  the  century  is 
divisible  by  4.     Thus  while  1600  remained  a  leap  year,  as 
before,  1700,  1800,  and  1900  were  to  be  common  years. 

This  change  in  the  calendar  was  speedily  adopted  by  all 


THE  CALENDAR.  255 

Catholic  countries,  and  more  slowly  by  Protestant  ones, 
England  holding  out  until  1752.  In  Kussia  it  has  never 
been  adopted  at  all,  the  Julian  calendar  being  still  con- 
tinued without  change.  The  Kussian  reckoning  is  there- 
fore 12  days  behind  ours,  the  ten  days  dropped  in  1582 
being  increased  by  the  days  dropped  from  the  years  1700 
and  1800  in  the  new  reckoning.  This  modified  calendar 
is  called  the  Gregorian  Calendar,  or  New  Style,  while  the 
old  system  is  called  the  Julian  Calendar,  or  Old  Style. 

It  is  to  be  remarked  that  the  practice  of  commencing 
the  year  on  January  1st  was  not  universal  until  compara- 
tively recent  times.  During  the  first  sixteen  centuries  of 
the  Julian  calendar  there  was  such  an  absence  of  definite 
rules  on  this  subject,  and  such  a  variety  of  practice  on  the 
part  of  different  powers,  that  the  simple  enumeration  of 
the  times  chosen  by  various  governments  and  pontiffs  for 
the  commencement  of  the  year  would  make  a  tedious 
chapter.  The  most  common  times  of  commencing  were, 
perhaps,  March  1st  and  March  22d,  the  latter  being  the 
time  of  the  vernal  equinox.  But  January  1st  gradually 
made  its  way,  and  became  universal  after  its  adoption  by 
England  in  1752. 

Solar  Cycle  and  Dominical  Letter. — In  our  church  cal- 
endars January  1st  is  marked  by  the  letter  A,  January  2d 
by  B,  and  so  on  to  G,  when  the  seven  letters  begin  over 
again,  and  are  repeated  through  the  year  in  the  same 
order.  Each  letter  there  indicates  the  same  day  of  the 
week  throughout  each  separate  year,  A  indicating  the  day 
on  which  January  1st  falls,  B  the  day  following,  and  so 
on.  An  exception  occurs  in  leap  years,  when  February 
29th  and  March  1st  are  marked  by  the  same  letter,  so  that 
a  change  occurs  at  the  beginning  of  March.  The  letter 
corresponding  to  Sunday  on  this  scheme  is  called  the  Do- 
minical or  Sunday  letter,  and,  when  we  once  know  what 
letter  it  is,  all  the  Sundays  of  the  year  are  indicated  by 
that  letter,  and  hence  all  the  other  days  of  the  week  by 
their  letters.  In  leap  years  there  will  be  two  Dominical 


256  ASTRONOMY. 

letters,  that  for  the  last  ten  months  of  the  year  being  the 
one  next  preceding  the  letter  for  January  and  February. 
In  the  Julian  calendar  the  Dominical  letter  must  always 
recur  at  the  end  of  28  years  (besides  three  recurrences  at 
unequal  intervals  in  the  mean  time).  This  period  is  called 
the  solar  cycle,  and  determines  the  days  of  the  week  on 
which  the  days  of  the  month  fall  during  each  year. 

Since  any  day  of  the  year  occurs  one  day  earlier  in  the 
week  than  it  did  the  year  before,  or  two  days  earlier  when 
a  29th  of  February  has  intervened,  the  Dominical  letters 
recur  in  the  order  G,  F,  E,  D,  C,  B,  A,  G,  etc.  A 
similar  fact  may  be  expressed  by  saying  that  any  day  of 
the  year  occurs  one  day  later  in  the  week  for  every  year 
that  has  elapsed,  and,  in  addition,  one  day  later  for  every 
29th  of  February  that  has  intervened.  This  fact  will  make 
it  easy  to  calculate  the  day  of  the  week  on  which  any  his- 
torical event  happened  from  the  day  corresponding  in  any 
past  or  future  year.  Let  us  take  the  following  example  : 

On  what  day  of  the  week  was  WASHINGTON  born,  the 
date  being  1732,  February  22d,  knowing  that  February 
22d,  1879,  fell  on  Saturday.  The  interval  is  147  years  : 
dividing  by  4  we  have  a  quotient  of  36  and  a  remainder 
of  3,  showing  that,  had  every  fourth  year  in  the  interval 
been  a  leap  year,  there  were  either  36  or  37  leap  years. 
As  a  February  29th  followed  only  a  week  after  the  date, 
the  number  must  be  37  ;*  but  as  1800  was  dropped  from 
the  list  of  leap  years,  the  number  was  really  only  36. 
Then  147  +  36  =  183  days  advanced  in  the  week.  Di- 
viding by  7,  because  the  same  day  of  the  week  recurs 
after  seven  days,  we  find  a  remainder  of  1.  So  February 
22d,  1879,  is  one  day  further  advanced  than  was  February 
22d,  1732  ;  so  the  former  being  Saturday,  WASHINGTON 
was  born  on  Friday. 

*  Perhaps  the  most  convenient  way  of  deciding  whether  the  remainder 
does  or  does  not  indicate  an  additional  leap  year  is  to  subtract  it  from  the 
last  date,  and  see  whether  a  February  29th  then  intervenes.  Subtract- 
ing 3  years  from  February  22d,  1879,  we  have  February  22d,  1876, 
and  a  29th  occurs  between  the  two  dates,  only  a  week  after  the  first. 


DIVISION  OF  THE  DAY  257 

§  3.    DIVISION  OP  THE  DAY. 

The  division  of  the  day  into  hours  was,  in  ancient  and 
mediaeval  times,  effected  in  a  way  very  different  from  that 
which  we  practice.  Artificial  time-keepers  not  being  in 
general  use,  the  two  fundamental  moments  were  sunrise 
and  sunset,  which  marked  the  day  as  distinct  from  the 
night.  The  first  subdivision  of  this  interval  was  marked 
by  the  instant  of  noon,  when  the  sun  was  on  the  meridian. 
The  day  was  thus  subdivided  into  two  parts.  The  night 
was  similarly  divided  by  the  times  of  rising  and  culmina- 
tion of  the  various  constellations.  EURIPIDES  (480-407 
B.C.)  makes  the  chorus  in  Rhesus  ask  : 

"  CHORUS.— Whose  is  the  guard  ?  Who  takes  my  turn  ?  The  first 
signs  are  setting,  and  the  seven  Pleiades  are  in  the  sky,  and  tJie  Eagle  glides 
midway  through  Jwaven.  Awake  !  Why  do  you  delay  ?  Awake  from 
your  beds  to  watch  !  See  ye  not  the  brilliancy  of  the  moon  ?  Morn, 
morn  indeed  is  approaching,  arid  hither  is  one  of  tlie  forerunning  stars." 
—The  Tragedies  of  Euripides.  Literally  Translated  by  T.  A.  Buckley. 
London  :  H.  G.  Bohn.  1854.  Vol.  2,  p.  322. 

The  interval  between  sunrise  and  sunset  was  divided 
into  twelve  equal  parts  called  hours,  and  as  this  interval 
varied  with  the  season,  the  length  of  the  hour  varied  also. 
The  night,  whether  long  or  short,  was  divided  into  hours 
of  the  same  character,  only,  when  the  night  hours  were 
long,  those  of  the  day  were  short,  and  vice  versa.  These 
variable  hours  were  called  temporary  hours.  At  the  time 
of  the  equinoxes,  both  the  day  and  the  night  hours  were 
of  the  same  length  with  those  we  use — namely,  the  twenty- 
fourth  part  of  the  day  ;  these  were  therefore  called  equi- 
noctial hours. 

The  use  of  these  temporary  hours  was  intimately  as- 
sociated with  the  time  of  beginning  of  the  day.  Instead 
of  commencing  the  civil  day  at  midnight,  as  we  do,  it  was 
customary  to  commence  it  at  sunset.  The  Jewish  Sabbath, 
for  instance,  commenced  as  soon  as  the  sun  set  on  Friday, 
and  ended  when  it  set  on  Saturday.  This  made  a  more 
distinctive  division  of  the  astronomical  day  than  that 


258  ASTRONOMY. 

which  we  employ,  and  led  naturally  to  considering  the 
day  and  the  night  as  two  distinct  periods,  each  to  be  di- 
vided into  12  hours. 

So  long  as  temporary  hours  were  used,  the  beginning  of 
the  day  and  the  beginning  of  the  night,  or,  as  we  should 
call  it,  six  o'clock  in  the  morning  and  six  o'clock  in  the 
evening,  were  marked  by  the  rising  and  setting  of  the  sun  ; 
but  when  equinoctial  hours  were  introduced,  neither  sun- 
rise nor  sunset  could  be  taken  to  count  from,  because  both 
varied  too  much  in  the  course  of  the  year.  It  therefore 
became  customary  to  count  from  noon,  or  the  time  at 
which  the  sun  passed  the  meridian.  The  old  custom  of 
dividing  the  day  and  the  night  each  into  12  parts  was  con- 
tinued, the  first  12  being  reckoned  from  midnight  to 
noon,  and  the  second  from  noon  to  midnight.  The  day 
was  made  to  commence  at  midnight  rather  than  at  noon 
for  obvious  reasons  of  convenience,  although  noon  was  of 
course  the  point  at  which  the  time  had  to  be  determined. 

Equation  of  Time. — To  any  one  who  studied  the  annual 
motion  of  the  sun,  it  must  have  been  quite  evident  that 
the  intervals  between  its  successive  passages  over  the 
meridian,  or  between  one  noon  and  the  next,  could  not 
be  the  same  throughout  the  year,  because  the  apparent 
motion  of  the  sun  in  right  ascension  is  not  constant.  It 
will  be  remembered  that  the  apparent  revolution  of  the 
starry  sphere,  or,  which  is  the  same  thing,  the  diurnal 
revolution  of  the  earth  upon  its  axis,  may  be  regarded 
as  absolutely  constant  for  all  practical  purposes.  This  rev- 
olution is  measured  around  in  right  ascension  as  explained 
in  the  opening  chapter  of  this  work.  If  the  sun  increased 
its  right  ascension  by  the  same  amount  every  day,  it  would 
pass  the  meridian  3'  56"  later  every  day,  as  measured  by 
sidereal  time,  and  hence  the  intervals  between  successive 
passages  would  be  equal.  But  the  motion  of  the  sun  in 
right  ascension  is  unequal  from  two  causes  :  (1)  the  un- 
equal motion  of  the  earth  in  its  annual  revolution  around 
it,  arising  from  the  eccentricity  of  the  orbit,  and  (2)  the 


APPARENT  AND  MEAN  TIME.  259 

obliquity  of  the  ecliptic.  How  the  first  cause  produces  an 
inequality  is  obvious,  and  its  approximate  amount  is  readily 
computed.  We  have  seen  that  the  angular  velocity  of  a 
planet  around  the  sun  is  inversely  as  the  square  of  its  ra- 
dius vector.  Taking  the  distance  of  the  earth  from  the  sun 
as  unity,  and  putting  e  for  the  eccentricity  of  its  orbit,  its 
greatest  distance  about  the  end  of  June  is  1  +  e  =  1-0168, 
and  its  least  distance  about  the  end  of  December  is 
1—0-0168.  The  squares  of  these  quantities  are  1  •  034  and 
1  —  034-  very  nearly  ;  therefore  the  motion  is  about  one 
thirtieth  greater  than  the  mean  in  December  and  one 
thirtieth  less  in  June.  The  mean  motion  is  3tn  563  ;  the 
actual  motion  therefore  varies  from  3m  48s  to  4m  4s. 

The  effect  of  the  obliquity  of  the  ecliptic  is  still  greater. 
When  the  sun  is  near  the  equinox,  its  motion  along  the 
ecliptic  makes  an  angle  of  23J°  with  the  parallels  of  dec- 
lination. Since  its  motion  in  right  ascension  is  reckoned 
along  the  parallel  of  declination,  we  see  that  it  is  equal  to 
the  motion  in  longitude  multiplied  by  the  cosine  of  23-J0. 
This  cosine  is  less  than  unity  by  about  -07  ;  therefore 
at  the  times  of  the  equinox  the  mean  motion  is  diminished 
by  this  fraction,  or  by  20  seconds.  Therefore  the  days 
are  then  20  seconds  shorter  than  they  would  be  were  there 
no  obliquity.  At  the  solstices  the  opposite  effect  is  pro- 
duced. Here  the  different  meridians  of  right  ascension 
are  nearer  together  than  they  are  at  the  equator  in  the 
proportion  of  the  cosine  of  23J°  to  unity  ;  therefore,  when 
the  sun  moves  through  one  degree  along  the  ecliptic,  it 
changes  its  right  ascension  by  1  •  08°  ;  here,  therefore,  the 
days  are  about  19  seconds  longer  than  they  would  be  if  the 
obliquity  of  the  ecliptic  was  zero. 

We  thus  have  to  recognize  two  slightly  different  kinds 
of  days  :  solar  days  and  mean  days.  A  solar  day  is  the 
interval  of  time  between  two  successive  transits  of  the  sun 
over  the  same  meridian,  while  a  mean  day  is  the  mean  of 
all  the  solar  days  in  a  year.  If  we  had  two  clocks,  the 
one  going  with  perfect  uniformity,  but  regulated  so  as  to 


260  ASTRONOMY. 

keep  as  near  the  sun  as  possible,  and  the  other  changing 
its  rate  so  as  to  always  follow  the  sun,  the  latter  would  gain 
or  lose  on  the  former  by  amounts  sometimes  rising  to  22 
seconds  i~i  a  day.  The  accumulation  of  these  variations 
through  a  period  of  several  months  would  lead  to  such 
deviations  that  the  sun-clock  would  be  14  minutes  slower 
than  the  other  during  the  first  half  of  February,  and  16 
minutes  faster  during  the  first  week  in  November.  The 
time-keepers  formerly  used  were  so  imperfect  that  these 
inequalities  in  the  solar  day  were  nearly  lost  in  the  neces- 
sary irregularities  of  the  rate  of  the  clock.  All  clocks 
were  therefore  set  by  the  sun  as  often  as  was  found  neces- 
sary or  convenient.  But  during  the  last  century  it  was 
found  by  astronomers  that  the  use  of  units  of  time  vary- 
ing in  this  way  led  to  much  inconvenience  ;  they  there- 
fore substituted  mean  time  for  solar  or  apparent  time. 

Mean  time  is  so  measured  that  the  hours  and  days  shall 
always  be  of  the  same  length,  and  shall,  on  the  average,  be 
as  much  behind  the  sun  as  ahead  of  it.  We  may  imagine 
a  fictitious  or  mean  sun  moving  along  the  equator  at  the 
rate  of  3m  56s  in  right  ascension  every  day.  Mean  time 
will  then  be  measured  by  the  passage  of  this  fictitious  sun 
across  the  meridian.  Apparent  time  was  used  in  ordinary 
life  after  it  was  given  up  by  astronomers,  because  it  was 
very  easy  to  set  a  clock  from  time  to  time  as  the  sun 
passed  a  noon-mark.  But  when  the  clock  was  so  far  im- 
proved that  it  kept  much  better  time  than  the  sun  did,  it 
was  found  troublesome  to  keep  putting  it  backward  and 
forward,  so  as  to  agree  with  the  sun.  Thus  mean  time 
was  gradually  introduced  for  all  the  purposes  of  ordinary 
life  except  in  very  remote  country  districts,  where  the 
farmers  may  find  it  more  troublesome  to  allow  for  an  equa- 
tion of  time  than  to  set  their  clocks  by  the  sun  every  few 
days. 

The  common  household  almanac  should  give  the  equa- 
tion of  time,  or  the  mean  time  at  which  the  sun  passes  the 
meridian,  on  each  day  of  the  year.  Then,  if  any  one  wishes 


IMPROVING   THE  CALENDAR.  261 

to  set  his  clock,  he  knows  the  moment  of  the  sun  passing 
the  meridian,  or  being  at  some  noon-mark,  and  sets  his 
time-piece  accordingly.  For  all  purposes  where  accurate 
time  is  required,  recourse  must  be  had  to  astronomical  ob- 
servation. It  is  now  customary  to  send  time-signals  every 
day  at  noon,  or  some  other  hour  agreed  upon,  from  obser- 
vatories along  the  principal  lines  of  telegraph.  Thus  at 
the  present  time  the  moment  of  Washington  noon  is  sig- 
nalled to  New  York,  and  over  the  principal  lines  of  rail- 
way to  the  South  and  West.  Each  person  within  reach  of 
a  telegraph-office  can  then  determine  his  local  time  by  cor- 
recting these  signals  for  the  difference  of  longitude. 

§  4.    REMARKS  ON  IMPROVING  THE  CALENDAR. 

It  is  an  interesting  question  whether  our  calendar,  this 
product  of  the  growth  of  ages,  which  we  have  so  rapidly 
described,  would  admit  of  decided  improvement  if  we 
were  frea  to  make  a  new  one  with  the  improved  materials 
of  modern  science.  This  question  is  not  to  be  hastily  an- 
swered in  the  affirmative.  Two  small  improvements  are 
undoubtedly  practicable  :  (1)  a  more  regular  division  of 
the  365  days  among  the  months,  giving  February  30  days, 
and  so  having  months  of  30  and  31  days  only  ;  (2)  putting 
the  additional  day  of  leap  year  at  the  end  of  the  year  in- 
stead of  at  the  end  of  February.  The  smallest  change 
from  our  present  system  would  be  made  by  taking  the  two 
additional  days  for  February,  the  one  from  the  end  of 
July,  and  the  other  from  the  end  of  December,  leaving 
the  last  with  30  days  in  common  years  and  31  in  leap 
years.  When  we  consider  more  radical  changes  than  this, 
we  find  advantages  set  off  by  disadvantages.  For  in- 
stance, it  would  on  some  accounts  be  very  convenient  to 
divide  the  year  into  13  months  of  4  weeks  each,  the  last 
month  having  one  or  two  extra  days.  The  months  would 
then  begin  on  the  same  day  of  the  week  through  each 
year,  and  would  admit  of  a  much  more  convenient  subdi- 


262  ASTRONOMY. 

vision  into  halves  and  quarters  than  they  do  now.  But  the 
year  would  not  admit  of  sucn  a  subdivision  without  divid- 
ing the  months  also,  and  it  is  possible  that  this  inconven- 
ience would  balance  the  conveniences  of  the  plan. 

An  actual  attempt  in  modern  times  to  form  an  entirely 
new  calendar  is  of  sufficient  historic  interest  to  be  men- 
tioned in  this  connection.  We  refer  to  the  so-called  Repub- 
lican Calendar  of  revolutionary  France.  The  year  some- 
times had  365  and  sometimes  366  days,  but  instead  of 
having  the  leap  years  at  defined  intervals,  one  was  inserted 
whenever  it  might  be  necessary  to  make  the  autumnal 
equinox  fall  on  the  first  day  of  the  year.  The  division  of 
the  year  was  effected  after  the  plan  of  the  ancient  Egyp- 
tians, there  being  12  months  of  30  days  each,  followed  by 
5  or  6  supplementary  days  to  complete  the  year,  which 
were  kept  as  feast-days.*  The  sixth  day  of  course  occur- 
red only  in  the  leap  years,  or  Franciads  as  they  were  call- 
ed. It  was  called  the  Day  of  the  Revolution,  and  was  set 
apart  for  a  quadrennial  oath  to  remain  free  or  die. 

No  attempt  was  made  to  fit  the  new  calendar  to  the  old 
one,  or  to  render  the  change  natural  or  convenient.  The 
year  began  with  the  autumnal  equinox,  or  September  22d 
of  the  Gregorian  calendar  ;  entirely  new  names  were 
given  to  the  months  ;  the  week  was  abolished,  and  in  lieu 
of  it  the  month  was  divided  into  three  decades,  the  last  or 
tenth  day  of  each  decade  being  a  holiday  set  apart  for  the 
adoration  of  some  sentiment.  Even  the  division  of  the  day 
into  24  hours  was  done  away  with,  arid  a  division  into 
ten  hours  was  substituted. 

The  Republican  Calendar  was  formed  in  1793,  the  year 
1  commencing  on  September  22d,  1792,  and  it  was 
abolished  on  January  1st,  1806,  after  13  years  of  con- 
fusion. 

*  They  received  the  nickname  of  sans-culottides,  from  the  opponents 
of  the  new  state  of  things. 


THE  ASTRONOMICAL   EPHEMERIS.  263 

§  5.    THE  ASTRONOMICAL    EPHEMERIS,  OR   NAU- 
TICAL ALMANAC. 

The  Astronomical  Ephemeris,  or,  as  it  is  more  com- 
monly called,  the  Nautical  Almanac,  is  a  work  in  which 
celestial  phenomena  and  the  positions  of  the  heavenly 
bodies  are  computed'in  advance.  The  need  of  such  a  work 
must  have  been  felt  by  navigators  and  astronomers  from 
the  time  that  astronomical  predictions  became  sufficiently 
accurate  to  enable  them  to  determine  their  position  on  the 
surface  of  the  earth.  At  first  works  of  this  class  were  pre- 
pared and  published  by  individual  astronomers  who  had 
the  taste  and  leisure  for  this  kind  of  labor.  MANFEEDI, 
of  Bonn,  published  Ephemerides  in  two  volumes,  which 
gave  the  principal  aspects  of  the  heavens,  the  positions  of 
the  stars,  planets,  etc.,  from  1715  until  1725.  This  work 
included  maps  of  the  civilized  world,  showing  the  paths  of 
the  principal  eclipses  during  this  interval. 

The  usefulness  of  such  a  work,  especially  to  the  naviga- 
tor, depends  upon  its  regular  appearance  on  a  uniform  plan 
and  upon  the  fulness  and  accuracy  of  its  data  ;  it  was  there- 
fore necessary  that  its  issue  should  be  taken  up  as  a  gov- 
ernment work.  Of  works  of  this  class  still  issued  the 
earliest  was  the  Connaissance  des  Temps  of  France,  the 
first  volume  of  which  -was  published  by  PICARD  in  1679, 
and  which  has  been  continued  without  interruption  until 
the  present  time.  The  publication  of  the  British  Nautical 
Almanac  was  commenced  in  the  year  1767  on  the  repre- 
sentations of  the  Astronomer  Royal  showing  that  such  a 
work  would  enable  the  navigator  to  determine  his  longi- 
tude within  one  degree  by  observations  of  the  moon.  An 
astronomical  or  nautical  almanac  is  now  published  annually 
by  each  of  the  governments  of  Germany,  Spain,  Portugal, 
France,  Great  Britain,  and  the  United  States.  They  have 
gradually  increased  in  size  and  extent  with  the  advancing 
wants  of  the  astronomer  until  those  of  Great  Britain  and 
this  country  have  become  octavo  volumes  of  between  500 


264  ASTRONOMY. 

and  600  pages.  These  two  are  published  three  years  or 
more  beforehand,  in  order  that  navigators  going  on  long 
voyages  may  supply  themselves  in  advance.  The  Ameri- 
can Ephemeris  and  Nautical  Almanae  has  been  regular- 
ly published  since  1855,  the  first  volume  being  for  that 
year.  It  is  designed  for  the  use  of  navigators  the  world 
over,  and  the  greater  part  of  it  is  especially  arranged  for 
the  use  of  astronomers  in  the  United  States. 

The  immediate  object  of  publications  of  this  class  is  to 
enable  the  wayfarer  and  traveller  upon  land  and  the  voy- 
ager upon  the  ocean  to  determine  their  positions  by  obser- 
vations of  the  heavenly  bodies.  Astronomical  instruments 
and  methods  of  calculation  have  been  brought  to  such  a 
degree  of  perfection  that  an  astronomer,  armed  with  a  nau- 
tical almanac,  u  chronometer  regulated  to  Greenwich  or 
Washington  time,  a  catalogue  of  stars,  and  the  necessary 
instruments  of  observation,  can  determine  his  position  at 
any  point  on  the  earth's  surface  within  a  hundred  yards 
by  a  single  night's  observations.  If  his  chronometer  is 
not  so  regulated,  he  can  still  determine  his  latitude,  but  not 
his  longitude.  He  could,  however,  obtain  a  rough  idea 
of  the  latter  by  observations  upon  the  planets,  and  come 
within  a  very  few  miles  of  it  by  a  single  observation  on 
the  moon. 

The  Ephemeris  furnishes  the  fundamental  data  from 
which  all  our  household  almanacs  are  calculated. 

The  principal  quantities  given  in  the  American  Ephemeris  for 
each  year  are  as  follows  : 

The  positions  of  the  sun  and  the  principal  large  planets  for  Green- 
wich noon  of  every  day  in  each  year. 

The  right  ascension  and  declination  of  the  moon's  centre  for 
every  hour  in  the  year. 

The  distance  of  the  moon  from  certain  bright  stars  and  planets 
for  every  third  hour  of  the  year. 

The  right  ascensions  and  declinations  of  upward  of  two  hundred 
of  the  brighter  fixed  stars,  corrected  for  precession,  nutation,  and 
aberration,  for  every  ten  days. 

The  positions  of  the  principal  planets  at  every  visible  transit  over 
the  meridian  of  Washington. 

Complete  elements  of  all  the  eclipses  of  the  sun  and  moon,  with 


THE  EPHEMERI8. 


265 


maps  showing  the  passage  of  the  moon's  shadow  or  penumbra  over 
those  regions  of  the  earth  where  the  eclipses  will  be  visible,  and 
tables  whereby  the  phases  of  the  eclipses  can  be  accurately  com- 
puted for  any  place. 

Tables  for  predicting  the  occultations  of  stars  by  the  moon. 

Eclipses  of  Jupiter's  satellites  and  miscellaneous  phenomena. 

To  give  the  reader  a  still  further  idea  of  the  Ephemeris,  we  pre- 
sent a  small  portion  of  one  of  its  pages  for  the  year  1882 : 

FEBRUARY,  1882 — AT  GREENWICH  MEAN  NOON. 


V 

THE  SUN'S 

Equation 

i* 
A 

5  • 

of  time  to 

1-1 

Sidereal  time 

Day  01 

££ 

be  sub- 

t- 
o 

or  right  as- 

the 
week. 

f  0 

£s 

Apparent 
right  ascen- 
sion. 

Diff. 
for  1 
hour. 

Apparent  de- 
clmatiou. 

Diff. 
forl 
hour. 

tracted 
from  mean 

time. 

P 

cension  of 
mean  mm. 

H.      M.           S. 

s. 

Of* 

. 

M.         S. 

s. 

H.     M.         8. 

Wed. 

1 

21      0    13-04 

10-175 

S  17      2    22-4 

+42-82 

13    51-34 

0-318 

20    46    21-70 

Thur. 

2 

21      4    16-84 

10-141 

16    45      5-4 

43-57 

13    58-58 

0-284 

20    50    18-26 

Frid. 

3 

21      8    19-82 

10-107 

16    27    30-9 

44-30 

14      5-01 

0-250 

20    54    14-81 

Sat. 

4 

21     12    21-98 

10  073 

16      9    39-2 

+44-99 

14    10-61 

0-216 

20    58    11-37 

Sun. 

5 

Z\     16    23-33 

10-040 

15    51    30-8 

45-69 

14    15-41 

o-ias 

21      2      7  92 

Mon. 

6 

21    20    23-88 

10-007 

15    33      6-1 

46-36 

14    19-40 

0-150 

21      6      4-48 

Tues. 

7 

21    24    23-63 

9-974 

15    14    25-4 

+47-03 

14    22-60 

0-117 

21    10      1-03 

Wed. 

8 

21    28    22-60 

9-941 

14    55    29-1 

47-66 

14    25-01 

0-084 

21    13    57-59 

Thur. 

9 

21    32    20-79 

9-909 

14    36    17-7 

48-28 

14    26-65 

0-052 

21     17    54-14 

Frid. 

10 

21     36    18-21 

9-877 

14    16    51-6 

48-88 

14    27-51 

0-020 

21    21    50-70 

Sat. 

11 

21    40    14-88 

9-846 

13    57    11-2 

49-47 

14    27-63 

0-011 

21    25    47-25 

Sun. 

12 

21    44    10-80 

9-815 

13    37    16-9 

50-03 

14    26-99 

0-042 

21    29    43-81 

Mon. 

13 

21    48      5-98 

9-784 

13    17      9-1 

450-59 

14    25-63 

0-073 

21    a3    40-35 

Tues. 

14 

21    52      0-43 

9-753 

12    56    48-3 

51-12 

14    23-52 

0-104 

21    37    36-91 

Wed. 

15 

21     55    54-16 

9-723 

12    36    14-9 

51-65 

14    20-70 

0-134 

21    41    33-46 

Thur. 

16 

21    59    47-17 

9-693 

12    15    29-3 

+52-14 

14    17-15 

0-164 

21    45    30-02 

Frid. 

17    22      3    39-47 

9-664 

11    54    32-1 

52-62 

14    12-90 

0-193 

21    49    26-57 

Sat. 

18 

22      7    31-07 

9-635 

11    33    23-6 

53-07 

14      7-94 

0-222 

21    53    23-13 

Of  the  same  general  nature  with  the  Ephemeris  are  catalogues  of 
the  fixed  stars.  The  object  of  such  a  catalogue  is  to  give  the  right 
ascension  and  declination  of  a  number  of  stars  for  some  epoch,  the 
beginning  of  the  year  1875  for  instance,  with  the  data  by  which  the 
position  of  a  star  can  be  found  at  any  other  epoch.  Such  cata- 
logues are,  however,  imperfect  owing  to  the  constant  small  changes 
in  the  positions  of  the  stars  and  the  errors  and  imperfections  of  the 
older  observations.  In  consequence  of  these  imperfections,  a  consid- 
erable part  of  the  work  of  the  astronomer  engaged  in  accurate  de- 
terminations of  geographical  positions  consist  in  finding  the  most 
accurate  positions  of  the  stars  which  he  makes  use  of. 


PART   II. 

THE  SOLAR  SYSTEM  IN  DETAIL. 


CHAPTER   I. 

STRUCTURE  OF  THE  SOLAR  SYSTEM. 

THE  solar  system,  as  it  is  known  to  us  through  the  dis- 
coveries of  COPERNICUS,  KEPLER,  NEWTON  and  their  suc- 
cessors, consists  of  the  sun  as  a  central  body,  around  which 
revolve  the  major  and  minor  planets,  with  their  satellites, 
a  few  periodic  comets,  and  an  unknown  number  of  meteor 
swarms.  These  are  permanent  members  of  the  system. 
At  times  other  comets  appear,  and  move  usually  in  par- 
abolas through  the  system,  around  the  sun,  and  away  from 
it  into  space  again,  thus  visiting  the  system  without  be- 
ing permanent  members  of  it. 

The  bodies  of  the  system  may  be  classified  as  follows  : 

1.  The  central  body — the  Sun. 

2.  The  four  inner  planets — Mercury,  Venus,  the  Earth, 
Mars. 

3.  A  group  of  small  planets,  sometimes  called  Asteroids, 
revolving  outside  of  the  orbit  of  Mars. 

4.  A  group  of  four  outer  planets — Jupiter,   Saturn, 
Uranus,  and  Neptune. 

5.  The  satellites,  or  secondary  bodies,  revolving  about 
the  planets,  their  primaries. 

6.  A  number  of  comets  and  meteor  swarms  revolving 
in  very  eccentric  orbits  about  the  Sun. 


268  ASTRONOMY. 

The  eight  planets  of  Groups  2  and  4  are  sometimes 
classed  together  as  the  major  planets,  to  distinguish  them 
from  the  two  hundred  or  more  minor  planets  of  Group  3. 
The  formal  definitions  of  the  various  classes,  laid  down 
by  Sir  WILLIAM  HERSCHEL  in  1802,  are  worthy  of  repe- 
tition : 

Planets  are  celestial  bodies  of  a  certain  very  consider- 
able size. 

They  move  in  not  very  eccentric  ellipses  about  the 
sun. 

The  planes  of  their  orbits  do  not  deviate  many  degrees 
from  the  plane  of  the  earth's  orbit. 

Their  motion  about  the  sun  is  direct. 

They  may  have  satellites  or  rings. 

They  have  atmospheres  of  considerable  extent,  which, 
however,  bears  hardly  any  sensible  proportion  to  their 
diameters. 

Their  orbits  are  at  certain  considerable  distances  from 
each  other. 

Asteroids,  now  more  generally  known  as  small  or 
minor  planets,  are  celestial  bodies  which  move  about  the 
sun  in  orbits,  either  of  little  or  of  considerable  eccen- 
tricity, the  planes  of  which  orbits  may  be  inclined  to  the 
ecliptic  in  any  angle  whatsoever.  They  may  or  may  not 
have  considerable  atmospheres. 

Comets  are  celestial  bodies,  generally  of  a  very  small 
mass,  though  how  far  this  may  be  limited  is  yet  un- 
known. 

They  move  in  very  eccentric  ellipses  or  in  parabolic 
arcs  about  the  sun. 

The  planes  of  their  motion  admit  of  the  greatest  variety 
in  their  situation. 

The  direction  of  their  motion  is  also  totally  undeter- 
mined. 

They  have  atmospheres  of  very  great  extent,  which 
show  themselves  in  various  forms  as  tails,  coma,  haziness, 
etc. 


MAGNITUDES  OF  THE  PLANETS. 


269 


Relative  Sizes  of  the  Planets.— The  comparative  sizes  of 
the  major  planets,  as  they  would  appear  to  an  observer 
situated  at  an  equal  distance  from  all  of  them,  is  given  in 
the  following  figure. 


5«IG>    74. — RELATIVE  SIZES  OF  THE  PLANETS. 

The  relative  apparent  magnitudes  of  the  sun,  as  seen 
from  the  various  planets,  is  shown  in  the  next  figure. 

Flora  and  Mnemosyne  are  two  of  the  asteroids. 

A  curious  relation  between  the  distances  of  the  planets, 
known  as  BODE'S  law,  deserves  mention.  If  to  the  num- 
bers, 

0,  3,  6,  12,  24,  48,  96,  192,  384, 


270  ASTRONOMY. 

each  of  which  (the  second  excepted)  is  twice  the  preced- 
ing, we  add  4,  we  obtain  tlie  series, 

4,  Y,  10,  16,  28,  52,  100,  196,  388. 
These  last  numbers  represent  approximately  the  dis- 


FlG.    75. — APPARENT  MAGNITUDES    OP    THE   SUN  AS    SEEN   FROM    DIF- 
FERENT PLANETS. 

tances  of  the  planets  from  the  sun  (except  for  Neptune, 
which  was  not  discovered  when  the  so-called  law  was  an- 
nounced). 

This  is  shown  in  the  following  table  : 


CHARACTERISTICS  OF  THE  PLANETS. 


PLANETS. 

Actual 
Distance. 

Bode's  Law. 

3-9 

4-0 

Venus                                      .  '.      .  . 

7-2 

7-0 

Earth     

10-0 

10-0 

Mars    

15-2 

16-0 

[Ceres] 

27-7 

28-0 

Jupiter                       .      .            

52-0 

52-0 

Saturn  

95-4 

100.0 

Uranus  

191-8 

196-0 

Neptune 

300-4 

388-0 

It  will  be  observed  that  Neptune  does  not  fall  within 
this  ingenious  scheme.  Ceres  is  one  of  the  minor  planets. 

The  relative  brightness  of  the  sun  and  the  various 
planets  has  been  measured  by  ZOLLNEK,  and  the  results 
are  given  below.  The  column  per  cent  shows  the  per- 
centage of  error  indicated  in  the  separate  results  : 


SUN  AND 

Ratio  :  1  to 

Percent,  of  Error. 

Moon  

618,000 

1.6 

Mars 

6  994  000  000 

5-8 

Jupiter     .         . 

5  472  000  000 

5-7 

Saturn  (ball  alone)  

130  980,000,000 

5-0 

Uranus 

8  486  000  000  000 

6-0 

Neptune 

79  620  000  000  000 

5-5 

The  differences  in  the  density,  size,  mass  and  distance 
of  the  several  planets,  and  in  the  amount  of  solar  light 
and  heat  which  they  receive,  are  immense.  The  distance 
of  Neptune  is  eighty  times  that  of  Mercury,  and  it  re- 
ceives only  ^O-Q  as  much  light  and  heat  from  the  sun. 
The  density  of  the  earth  is  about  six  times  that  of  water, 
while  Saturn's  mean  density  is  less  than  that  of  water. 

The  mass  of  the  sun  is  far  greater  than  that  of  any 
single  planet  in  the  system,  or  indeed  than  the  combined 
mass  of  all  of  them.  In  general,  it  is  a  remarkable  fact 
that  the  mass  of  any  given  planet  exceeds  the  sum  of  the 
masses  of  all  the  planets  of  less  mass  than  itself.  This  is 


272 


ASTRONOMY. 


shown  in  the  following  table,  where  the  masses  of  the  plan- 
ets are  taken  as  fractions  of  "the  sun's  mass,  which  we  here 
express  as  1,000,000,000: 


. 

oJ 

1 

E 

1 

£ 

00 

3 

C 

0 

1 

t~4 

3 

d 

PLANETS. 

* 

3 

fl 

P 

fe 

1 

* 

0} 

200 

334 

2,353 

3,060 

44,250 

51,600 

285,580, 

954,305 

1,000,000,000 

Masses. 

The  mass  of  Mercury  is  less  than  the  mass ) 
of  Mars :  \ 

The  sum  of  masses  of  Mercury  and  Mars ) 
is  less  than  the  mass  of  Venus  :  j" 

Mercury  +  Mars  +  Venus  <  Earth  : 

Mercury  +  Mars  +  Venus  4-  Earth  <  Ura- 
nus : 


Mercury  4-  Mars  4-  Venus  -f-  Earth 
nus  <  Neptune : 


Ura- 


200  < 

524  < 

2,877  < 

5,937  < 

50,187  < 


324 

2,353 

3,060 

44,250 


Mercury  -f-  Mars  4-  Venus  4-  Earth  -f-  Ura- 


nus +  Neptune  <  Saturn  :  101,787   < 


51,600 

285,580 

Mercury  +  Mars  +  Venus  4-  Earth  4-  Ura-  )      qft7  oP7     .  Qr4  OA* 

nus  4-  Neptune  +  Saturn  <  Jupiter  :      f  °4'dUl 

Combined  mass  of  all  the  planets  is  less  )    1  q41  fi79    .  ,  noft  ftnft  ftnn 
than  that  of  the  Sun  :  \  I,d41,b7^   <     1,000,000,00! 

The  total  mass  of  the  small  planets,  like  their  number, 
is  unknown,  but  it  is  probably  less  than  one  thousandth 
that  of  our  earth,  and  would  hardly  increase  the  sum  total 
of  the  above  masses  -of  the  solar  system  by  more  than  one 
or  two  units.  The  sun's  mass  is  thus  over  700  times  that 
of  all  the  other  bodies,  and  hence  the  fact  of  its  central 
position  in  the  solar  system  is  explained.  In  fact,  the 
centre  of  gravity  of  the  whole  solar  system  is  very  little 
outside  the  body  of  the  sun,  and  will  be  inside  of  it  when 
Jupiter  and  Saturn  are  in  opposite  directions  from  it. 

Planetary  Aspects. — The  motions  of  the  planets  about 
the  sun  have  been  explained  in  Chapter  IY.  From  what 
is  there  said  it  appears  that  the  best  time  to  see  one  of  the 


PLANETARY  ASPECTS.  273 

outer  planets  will  be  when  it  is  in  opposition — that  is,  when 
its  geocentric  longitude  or  its  right  ascension  differs  180° 
or  12h  from  that  of  the  sun.  At  such  a  time  the  planet 
will  rise  at  sunset  and  culminate  at  midnight.  During  the 
three  months  following  opposition,  the  planet  will  rise  from 
three  to  six  minutes  earlier  every  day,  so  that,  knowing 
when  a  planet  is  in  opposition,  it  is  easy  to  find  it  at  any 
other  time.  For  example,  a  month  after  opposition  the 


FIG.  76. 

planet  will  be  two  to  three  hours  high  about  sunset,  and 
will  culminate  about  nine  or  ten  o'clock.  Of  course  the 
inner  planets  never  come  into  opposition,  and  hence  are 
best  seen  about  the  times  of  their  greatest  elongations. 

The  above  figure  gives  a  rough  plan  of  part  of  the 
solar  system  as  it  would  appear  to  a  spectator  immediately 
above  or  below  the  plane  of  the  ecliptic. 


274  ASTRONOMY. 

It  is  drawn  approximately  to  scale,  the  mean  distance  of 
the  earth  (=  1)  being  half  an  inch.  The  mean  distance  of 
Saturn  would  be  4*77  inches,  of  Uranus  9*59  inches,  of 
Neptune  15  *03  inches.  On  the  same  scale  the  distance  of 
the  nearest  fixed  star  would  be  103, 133  inches,  or  over  one 
and  one  half  miles. 

The  arrangement  of  the  planets  and  satellites  is  then — 

The  Inner  Group.  Asteroids.                           The  Outer  Group. 

Mercury.  )     2()0      .         -I..-*-       (  Jupiter  and  4  moons. 

Venus.  ^UU  Tnor  P1J"Jet8«      )  Saturn  and  8  moons. 

Earth  and  Moon.  1  Uranus  and  4  moons. 

Mars  and  2  moons.  )                                          (  Neptune  and  1  moon. 

To  avoid  repetitions,  the  elements  of  the  major  planets 
and  other  data  are  collected  into  the  two  following  tables, 
to  which  reference  may  be  made  by  the  student.  The 
units  in  terms  of  which  the  various  quantities  are  given 
are  those  familiar  to  us,  as  miles,  days,  etc. ,  yet  some  of 
the  distances,  etc.,  are  so  immensely  greater  than  any 
known  to  our  daily  experience  that  we  must  have  recourse 
to  illustrations  to  obtain  any  idea  of  them  at  all.  For  ex- 
ample, the  distance  of  the  sun  is  said  to  be  92J  million 
miles.  It  is  of  importance  that  some  idea  should  be  had 
of  this  distance,  as  it  is  the  unit,  in  terms  of  which  not 
only  the  distances  in  the  solar  system  are  expressed,  but 
which  serves  as  a  basis  for  measures  in  the  stellar  universe. 
Thus  when  we  say  that  the  distance  of  the  stars  is  over 
200,000  times  the  mean  distance  of  the  sun,  it  becomes 
necessary  to  see  if  some  conception  can  be  obtained  of  one 
factor  in  this.  Of  the  abstract  number,  92,500,000,  we 
have  no  conception.  It  is  far  too  great  for  us  to  have 
counted.  We  have  never  taken  in  at  one  view,  even 
a  million  similar  discrete  objects.  To  count  from  1  to 
200  requires,  with  very  rapid  counting,  60  seconds.  Sup- 
pose this  kept  up  for  a  day  without  intermission  ;  at  the 
end  we  should  have  counted  288,000,  which  is  about  ¥^¥ 
of  92,500,000.  Hence  over  10  months'  uninterrupted 
counting  by  night  and  day  would  be  required  simply  to 
enumerate  the  number,  and  long  before  the  expiration  of 


EXTENT  OF  THE  SOLAR  SYSTEM.  275 

the  task  all  idea  of  it  would  have  vanished.  We  may  take 
other  and  perhaps  more  striking  examples.  We  know, 
for  instance,  that  the  time  of  the  fastest  express-trains  be- 
tween New  York  and  Chicago,  which  average  40  miles  per 
hour,  is  about  a  day.  Suppose  such  a  train  to  start  for 
the  sun  and  to  continue  running  at  this  rapid  rate.  It 
would  take  363  years  for  the  journey.  Three  hundred 
and  sixty- three  years  ago  there  was  not  a  European  settle- 
ment in  America. 

A  cannon-ball  moving  continuously  across  the  interven- 
ing space  at  its  highest  speed  would  require  about  nine 
years  to  reach  the  sun.  The  report  of  the  cannon,  if  it 
could  be  conveyed  to  the  sun  with  the  velocity  of  sound  in 
air,  would  arrive  there  five  years  after  the  projectile. 
Such  a  distance  is  entirely  inconceivable,  and  yet  it  is 
only  a  small  fraction  of  those  with  which  astronomy  has  to 
deal,  even  in  our  own  system.  The  distance  of  Neptune 
is  30  times  as  great. 

If  we  examine  the  dimensions  of  the  various  orbs,  we  meet 
almost  equally  inconceivable  numbers.  The  diameter 
of  the  sun  is  860,000  miles  ;  its  radius  is  but  430,000,  and 
yet  this  is  nearly  twice  the  mean  distance  of  the  moon 
from  the  earth.  Try  to  conceive,  in  looking  at  the  moon 
in  a  clear  sky,  that  if  the  centre  of  the  sun  could  be 
placed  at  the  centre  of  the  earth,  the  moon  would  be  far 
within  the  sun's  surface.  Or  again,  conceive  of  the  force 
of  gravity  at  the  surface  of  the  various  bodies  of  the  sys- 
tem. At  the  sun  it  is  nearly  28  times  that  known  to  us. 
A  pendulum  beating  seconds  here  would,  if  transported 
to  the  sun,  vibrate  with  a  motion  more  rapid  than  that  of 
a  watch-balance.  The  muscles  of  the  strongest  man  would 
not  support  him  erect  on  the  surface  of  the  sun  :  even 
lying  down  he  would  crush  himself  to  death  under  his 
own  weight  of  two  tons.  We  may  by  these  illustrations 
get  some  rough  idea  of  the  meaning  of  the  numbers  in 
these  tables,  and  of  the  incapability  of  our  limited  idea,s  to 
comprehend  the  true  dimensions  of  even  the  solar  system. 


276 


ASTRONOMY. 


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CHAPTER  II. 

THE    SUN. 
§   1.    GENERAL  SUMMARY. 

To  the  student  of  the  present  time,  armed  with  the 
powerful  means  of  research  devised  by  modern  science, 
the  sun  presents  phenomena  of  a  very  varied  and  complex 
character.  To  enable  the  nature  of  these  phenomena  to  be 
clearly  understood,  we  preface  our  account  of  the  physical 
constitution  of  the  sun  by  a  brief  summary  of  the  main 
features  seen  in  connection  with  that  body. 

Photosphere. — To  the  simple  vision  the  sun  presents 
the  aspect  of  a  brilliant  sphere.  The  visible  shining  sur- 
face of  this  sphere  is  called  the  photosphere,  to  distinguish 
it  from  the  body  of  the  sun  as  a  whole.  The  apparently 
flat  surface  presented  by  a  view  of  the  photosphere  is  called 
the  sun's  disk. 

Spots. — When  the  photosphere  is  examined  with  a  tele- 
scope, small  dark  patches  of  varied  and  irregular  outline 
are  frequently  found  upon  it.  These  are  called  the  solar 
spots. 

Rotation. — When  the  spots  are  observed  from  day  to 
day,  they  are  found  to  move  over  the  sun's  disk  in  such  a 
way  as  to  show  that  the  sun  rotates  on  its  axis  in  a  period 
of  25  or  26  days.  The  sun,  therefore,  has  axis,  poles,  and 
equator,  like  the  earth,  the  axis  being  the  line  around 
which  it  rotates. 

Faculse.— Groups  of  minute  specks  brighter  than  the 
general  surface  of  the  sun  are  often  seen  in  the  neighbor- 
hood of  spots  or  elsewhere.  They  are  called  faculce. 


FEATURES  OF  THE  SUN.  279 

Chromosphere,  or  Sierra. — The  solar  photosphere  is 
covered  by  a  layer  of  glowing  vapors  and  gases  of  very  ir- 
regular depth.  At  the  bottom  he  the  vapors  of  many 
metals,  iron,  etc.,  volatilized  by  the  fervent  heat  which 
reigns  there,  while  the  upper  portions  are  composed  prin- 
cipally of  hydrogen  gas.  This  vaporous  atmosphere  is 
commonly  called  the  chromosphere,  sometimes  the  sierra. 
It  is  entirely  invisible  to  direct  vision,  whether  with  the 
telescope  or  naked  eye,  except  for  a  few  seconds  about 
the  beginning  or  end  of  a  total  eclipse,  but  it  may  be  seen 
on  any  clear  day  through  the  spectroscope. 

Prominences,  Protuberances,  or  Bed  Flames. — The 
gases  of  the  chromosphere  are  frequently  thrown  up  in 
irregular  masses  to  vast  heights  above  the  photosphere,  it 
may  be  500,000,  100,000,  or  even  200,000  kilometres. 
Like  the  chromosphere,  these  masses  have  to  be  studied 
with  the  spectroscope,  and  can  never  be  directly  seen  ex- 
cept when  the  sunlight  is  cut  off  by  the  intervention  of  the 
moon  during  a  total  eclipse.  They  are  then  seen  as  rose- 
colored  flames,  or  piles  of  bright  red  clouds  of  irregular 
and  fantastic  shapes.  They  are  now  usually  called  "  prom- 
inences" by  the  English,  and  "  protuberances"  by 
French  writers. 

Corona. — During  total  eclipses  the  sun  is  seen  to  be  en- 
veloped by  a  mass  of  soft  white  light,  much  fainter  than 
the  chromosphere,  and  extending  out  on  all  sides  far  be- 
yond the  highest  prominences.  It  is  brightest  around  the 
edge  of  the  sun,  and  fades  off  toward  its  outer  boundary, 
by  insensible  gradations.  This  halo  of  light  is  called  the 
corona,  and  is  a  very  striking  object  during  a  total  eclipse. 

§   2.    THE  PHOTOSPHERE. 

Aspect  and  Structure  of  the  Photosphere. — The  disk 
of  the  sun  is  circular  in  shape,  no  matter  what  side  of  the 
sun's  globe  is  turned  toward  us,  whence  it  follows  that  the 
sun  itself  is  a  sphere.  The  aspect  of  the  disk,  when 


280  ASTRONOMY. 

viewed  with  the  naked  eye,  or  with  a  telescope  of 
low  power,  is  that  of  a  uniform  bright,  shining  surface, 
hence  called  the  photosphere.  With  a  telescope  of 
higher  power  the  photosphere  is  seen  to  be  diversified 
with  groups  of  spots,  and  under  good  conditions  the 
whole  mass  has  a  mottled  or  curdled  appearance.  This 
mottling  is  caused  by  the  presence  of  cloud-like  forms, 
whose  outlines  though  faint  are  yet  distinguishable. 
The  background  is  also  covered  with  small  white  dots 
or  forms  still  smaller  than  the  clouds.  These  are  the 
"  rice-grains, "  so  called.  The  clouds  themselves  are 
composed  of  small,  intensely  bright  bodies,  irregularly 
distributed,  of  tolerably  definite  shapes,  which  seem  to  be 
suspended  in  or  superposed  on  a  darker  medium  or  back- 
ground. The  spaces  between  the  bright  dots  vary  in 
diameter  from  2"  to  4"  (about  1400  to  2800  kilome- 
tres). The  rice-grains  themselves  have  been  seen  to 
be  composed  of  smaller  granules,  sometimes  not  more 
than  0"-3  (135  miles)  in  diameter,  clustered  together. 
Thus  there  have  been  seen  at  least  three  orders  of 
aggregation  in  the  brighter  parts  of  the  photosphere  : 
the  larger  cloud -like  forms  ;  the  rice  grains  ;  and,  small- 
est of  all,  the  granules.  These  forms  have  been  studied 
with  the  telescope  by  SECCHI,  HUGGINS,  and  LANGLEY, 
and  their  relations  tolerably  well  made  out. 

In  the  Annuaire  of  the  Bureau  of  Longitudes  for  1878  (p.  C89), 
M.  JANSSEN  gives  an  account  of  his  recent  discovery  of  the  reticulated 
arrangement  of  the  solar  photosphere.  The  paper  is  accompanied 
by  a  photograph  of  the  appearances  described,  which  is  enlarged 
threefold.  Photographs  less  than  four  inches  in  diameter  cannot 
satisfactorily  show  such  details.  As  the  granulations  of  the  solar 
surface  are,  in  general,  not  greatly  larger  than  1"  or  2",  the  photo- 
graphic irradiation,  which  is  sometimes  20"  or  more,  may  completely 
obscure  their  characteristics.  This  difficulty  M.  JANSSEN  has  over- 
come by  enlarging  the  image  and  shortening  the  time  of  expos- 
ure. In  this  way  the  irradiation  is  diminished,  because  as  the  di- 
ameters increase,  the  linear  dimensions  of  the  details  are  increased, 
and  "  the  imperfections  of  the  sensitive  plate  have  less  relative  im- 
portance." 


THE  SUN'S  PHOTOSPHERE. 


281 


Again,  M.  JANSSEN  has  noted  that  in  short  exposures  the  photo- 
graphic spectrum  is  almost  monochromatic. 

In  this  way  it  differs  greatly  from  the  visible  spectrum,  and  to 
the  advantage  of  the  former  for  this  special  purpose.  The  diameter 
of  the  solar  photograms  have  since  1874  been  successively  increased 
to  12,  15,  20,  and  30  centimetres.  The  exposure  is  made  equal  all 
over  the  surface.  In  summer  this  exposure  for  the  largest  photo- 


FlG.  77. — RETICULATED  ARRANGEMENT  OP  THE  PHOTOSPHERE. 

grams  is  less  than  09>0005.     The  development  of  such  pictures  is 
very  slow. 

These  photograms,  on  examination,  show  that  the  solar  surface  is 
covered  with  a  fine  granulation.  The  forms  and  the  dimensions  of 
the  elementary  surfaces  are  very  various.  They  vary  in  size  from 
0"-3  or  0"-4  to  3''  or  4"  (200  to  3000  kilometres).  Their  forms 


282  ASTRONOMY. 

are  generally  circles  or  ellipses,  but  these  curves  are  sometimes 
greatly  altered.  This  granulation  is  apparently  spread  equally  all 
over  the  disk.  The  brilliancy  of  the  points  is  very  variable,  and 
they  appear  to  be  situated  at  different  depths  below  the  photo- 
sphere :  the  most  luminous  particles,  those  to  which  the  solar  light 
is  chiefly  due,  occupy  only  a  small  fraction  of  the  solar  surface. 

The  most  remarkable  feature,  however,  is  ' '  the  reticulated  ar- 
rangement of  the  parts  of  the  photosphere."  "The  photograms 
show  that  the  constitution  of  the  photosphere  is  not  uniform 
throughout,  but  that  it  is  divided  in  a  series  of  regions  more  or 
less  distant  from  each  other,  and  having  each  a  special  constitution. 
These  regions  have,  in  general,  rounded  contours,  but  these  are 
often  almost  rectilinear,  thus  forming  polygons.  The  dimensions 
of  these  figures  are  very  variable  ;  some  are  even  1'  in  diameter 
(over  25,000  miles)."  "Between  these  figures  the  grains  are 
sharply  defined,  but  in  their  interior  they  ara  almost  effaced  and 
run  together  as  if  by  some  force."  These  phenomena  can  be  best 
understood  by  a  reference  to  the  figure  of  M.  JANSSEN  (p.  281). 

Light  and  Heat  from  the  Photosphere. — Tlie  photo- 
sphere is  not  equally  bright  all  over  the  apparent  disk. 
This  is  at  once  evident  to  the  eye  in  observing  the  sun  with 
a  telescope.  The  centre  of  the  disk  is  most  brilliant,  and 
the  edges  or  limbs  are  shaded  off  so  as  to  forcibly  suggest 
the  idea  of  an  absorptive  atmosphere,  which,  in  fact,  is  the 
cause  of  this  appearance. 

Such  absorption  occurs  not  only  for  the  rays  by  which 
we  see  the  sun,  the  so-called  visual  rays,  but  for  those 
which  have  the  most  powerful  effect  in  decomposing  the 
salts  of  silver,  the  so-called  chemical  rays,  by  which  the 
ordinary  photograph  is  taken. 

The  amount  of  heat  received  from  different  portions  of 
the  sun's  disk  is  also  variable,  according  to  the  part  of 
the  apparent  disk  examined.  This  is  what  we  should  ex- 
pect. That  is,  if  the  intensity  of  any  one  of  these  radiations 
(as  felt  at  the  earth)  varies  from  centre  to  circumference, 
that  of  every  other  should  also  vary,  since  they  are  all 
modifications  of  the  same  primitive  motion  of  the  sun's 
constituent  particles.  But  the  constitution  of  the  sun's 
atmosphere  is  such  that  the  law  of  variation  for  the  three 
classes  is  different.  The  intensity  of  the  radiation  in  the 
sun  itself  and  inside  of  the  absorptive  atmosphere  is  prob- 


SOLAR  RADIATION. 


283 


ably  nearly  constant.  The  ray  which  leaves  the  centre  of 
the  sun's  disk  in  passing  to  the  earth,  passes  through  the 
smallest  possible  thickness  of  the  solar  atmosphere,  while 
the  rays  from  points  of  the  sun's  body  which  appear  to 
us  near  the  limbs  pass,  on  the  contrary,  through  the  maxi- 
mum thickness  of  atmosphere,  and  are  thus  longest  sub- 
jected to  its  absorptive  action. 

This  is  plainly  a  rational  explanation,  since  the  part  of 
the  sun  which  is  seen  by  us  as  the  limb  varies  with  the 
position  of  the  earth  in  its  orbit  and  with  the  position  of 
the  sun's  surface  in  its  rotation,  and  has  itself  no  physical 
peculiarity.  The  various  absorptions  of  different  classes 
of  rays  correspond  to  this  supposition,  the  more  refrangi- 
ble rays  suffering  most  absorption,  as  they  must  do,  being 
composed  of  waves  of  shorter  wave  length. 

The  following  table  gives  the  observed  ratios  of  the  amount  of 
heat,  light,  and  chemical  action  at  the  centre  of  the  sun  and  at 
various  distances  from  the  centre  toward  the  limb.  The  first 
column  of  the  table  gives  the  apparent  distances  from  the  centre 
of  the  disk,  the  sun's  radius  being  1-00.  The  second  column  gives 
the  percentage  of  heat-rays  received  by  an  observer  on  the  earth 
from  points  at  these  various  distances.  That  is,  for  every  100  heat- 
rays  reaching  the  earth  from  the  sun's  centre,  95  reach  us  from  a 
point  half  way  from  the  centre  to  the  limb,  and  so  on. 

Analogous  data  are  given  for  the  light-rays  and  the  chemical 
rays.  The  data  in  regard  to  heat  are  due  to  Professor  LANGLEY  ; 
those  in  regard  to  light  and  chemical  action  to  Professor  PICKERING 
and  Dr.  VOGEL  respectively. 


DISTANCE  FROM 
CENTRE. 

Heat  Rays. 

Light  Rays. 

Chemical  Rays. 

0-00.. 
0-25  

100 
99 

100 
97 

100 
98 

0-50 

95 

91 

90 

0-75 

86 

79 

66 

0-85  

69 

48 

0-95  

55 

25 

0-96 

62 

23 

0-98  
1-00  

50 

37 

18 
13 

For  two  equal  apparent  surfaces,  A  near  the  sun's  centre  and  B 
near  the  limb,  we  may  say  that  the  rays  from  the  two  surfaces  when 


284  ASTRONOMY. 

received  at  the  earth  have  approximately  the  following  relative 
effects  : 

A  has  twice  as  much  effect  on  a  thermometer  as  B  (heat); 

A  has  three  times  as  much  illuminating  effect  as  B  (light); 

A  has  seven  times  as  much  effect  in  decomposing  the  photo- 
graphic salts  of  silver  as  B  (actinic  effect). 

It  is  to  be  carefully  borne  in  mind  that  the  above  numbers  refer 
to  variations  of  the  sun's  rays  received  from  different  equal  surfaces 
A  and  B,  in  their  effect  upon  certain  arbitrary  terrestrial  standards  of 
measure.  If,  for  example,  the  decomposition  of  other  salts  than 
those  employed  for  ordinary  photographic  work  be  taken  as  stand- 
ards, then  the  numbers  will  be  altered,  and  so  on.  We  are  simply 
measuring  the  power  of  solar  rays  selected  from  different  parts  of 
the  sun's  apparent  disk,  and  hence  exposed  to  different  conditions 
of  absorption  in  his  atmosphere,  to  do  work  of  a  certain  selected 
kind,  as  to  raise  the  temperature  of  a  thermometer,  to  affect  the 
human  retina,  or  to  decompose  certain  salts  of  silver. 

In  this  the  absorption  of  the  earth's  atmosphere  is  rendered  con- 
stant for  each  kind  of  experiment.  This  atmosphere  has,  however, 
a  very  strong  absorptive  effect.  We  know  that  we  can  look  at  the 
setting  or  rising  sun,  which  sends  its  light  rays  through  great 
depths  of  the  earth's  atmosphere,  but  not  upon  the  sun  at  noon- 
day. The  temperature  is  lower  at  sunrise  or  at  sunset  than  at  noon, 
and  the  absorption  of  chemical  rays  is  so  marked  that  a  photograph 
of  the  solar  spectrum  which  can  be  taken  in  three  seconds  at  noon 
requires  six  hundred  seconds  about  sunset — that  is,  two  hundred 
times  as  long  (DRAPER). 

Amount  of  Heat  Emitted  by  the  Sun. — Owing  to  the 
absorption  of  the  solar  atmosphere,  it  follows  that  we  re- 
ceive only  a  portion — perhaps  a  very  small  portion  —  of 
the  rays  emitted  by  the  sun's  photosphere. 

If  the  sun  had  no  absorptive  atmosphere,  it  would  seem 
to  us  hotter,  brighter,  and  more  blue  in  color. 

Exact  notions  as  to  how  great  this  absorption  is  are  hard 
to  gain,  but  it  may  be  said  roughly  that  the  best  authori- 
ties agree  that  although  it  is  quite  possible  that  the  sun's 
atmosphere  absorbs  half  the  emitted  rays,  it  probably  does 
not  absorb  four  fifths  of  them. 

It  is  a  curious,  and  as  yet  we  believe  unexplained  fact, 
that  the  absorption  of  the  solar  atmosphere  does  not  affect 
the  darkness  of  the  Fraunhofer  lines.  They  seem  equally 
black  at  the  centre  and  edge  of  the  sun.*  The  amount 

*  Prof.  YOUNG  has  spoken  of  a  slight  observable  difference. 


HEAT  OF  THE  SUtf.  285 

of  this  absorption  is  a  practical  question  to  us  on  the  earth. 
So  long  as  the  central  body  of  the  sun  continues  to  emit 
the  same  quantity  of  rays,  it  is  plain  that  the  thickness  of 
the  solar  atmosphere  determines  the  number  of  such  rays 
reaching  the  earth.  If  in  former  times  this  atmosphere 
was  much  thicker,  then  less  heat  would  have  reached  the 
earth.  Professor  LANGLEY  suggests  that  the  glacial  epoch 
may  be  explained  in  this  way.  If  the  central  body  of  the 
sun  has  likewise  had  different  emissive  powers  at  different 
times,  this  again  would  produce  a  variation  in  the  tempera- 
ture of  the  earth. 

Amount  of  Heat  Radiated. — There  is  at  present  no  way 
of  determining  accurately  either  the  absolute  amount  of 
heat  emitted  from  the  central  body  or  the  amount  of  this 
heat  stopped  by  the  solar  atmosphere  itself.  All  that  can 
be  done  is  to  measure  (and  that  only  roughly)  the  amount 
of  heat  really  received  by  the  earth,  without  attempting  to 
define  accurately  the  circumstances  which  this  radiation 
has  undergone  before  reaching  the  earth. 

The  difficulties  in  the  way  of  determining  how  much 
heat  reaches  the  earth  in  any  definite  time,  as  a  year,  are 
twofold.  First,  we  must  be  able  to  distinguish  between 
the  heat  as  received  by  a  thermometric  apparatus  from 
the  sun  itself  and  that  from  external  objects,  as  our  own 
atmosphere,  adjacent  buildings,  etc.;  and,  second,  we 
must  be  able  to  allow  for  the  absorption  of  the  earth's 
atmosphere. 

POUILLET  has  experimented  upon  this  question,  making 
allowance  for  the  time  that  the  sun  is  below  the  horizon 
of  any  place,  and  for  the  fact  that  the  solar  rays  do  not  in 
general  strike  perpendicularly  but  obliquely  upon  any 
given  part  of  the  earth's  surface.  His  conclusions  may 
be  stated  as  follows  :  if  our  own  atmosphere  were  re- 
moved, the  solar  rays  would  have  energy  enough  to  melt 
a  layer  of  ice  9  centimetres  thick  over  the  whole  earth 
daily,  or  a  layer  of  about  32  metres  thick  in  a  year. 

Of  the  total  amount  of  heat  radiated  by  the  sun,  the 


286  ASTRONOMY. 

earth  receives  but  an  insignificant  share.  The  sun  is 
capable  of  heating  the  entire*  surface  of  a  sphere  whose  ra- 
dius is  the  earth's  mean  distance  to  the  same  degree  that 
the  earth  is  now  heated.  The  surface  of  such  a  sphere  is 
2,170,000,000  times  greater  than  the  angular  dimensions 
of  the  earth  as  seen  from  the  sun,  and  hence  the  earth  re- 
ceives less  than  one  two  billionth  part  of  the  solar  radia- 
tion. The  rest  of  the  solar  rays  are,  so  far  as  we  know, 
lost  in  space. 

It  is  found,  from  direct  measures,  that  a  sun-spot  gives  less  heat, 
area  for  area,  than  the  unspotted  photosphere,  and  it  is  an  interest- 
ing question  how  much  the  climate  of  the  earth  can  be  affected  by 
this  difference. 

Professor  LANGLEY,  of  Pittsburgh,  has  made  measurements  of  the 
direct  effect  of  sun-spots  on  terrestrial  temperature.  The  observa- 
tions consisted  in  measuring  the  relative  amounts  of  umbral,  penum- 
bral,  and  photospheric  radiation.  The  relative  umbral,  penumbral, 
and  photospheric  areas  were  deduced  from  the  Kew  observations  of 
spots  ;  and  from  a  consideration  of  these  data,  and  confining  the 
question  strictly  to  changes  of  terrestrial  temperature  due  to  this 
cause  alone,  LANGLEY  deduces  the  result  that  "  sun-spots  do  ex- 
ercise a  direct  effect  on  terrestrial  temperature  by  decreasing  the 
mean  temperature  of  the  earth  at  their  maximum."  This  change 
is,  however,  very  small,  as  "it  is  represented  by  a  change  in  the 
mean  temperature  of  our  globe  in  eleven  years  not  greater  than 
0-3°  C.,  and  not  less  than  0-5°  C."  It  is  not  intended  to  show  that 
the  earth  is,  on  the  whole,  cooler  in  maximum  sun-spot  years,  but 
that,  as  far  as  this  cause  goes,  it  tends  to  make  the  earth  cooler  by 
this  minute  amount.  What  other  causes  may  co-exist  with  the 
maximum  spot-frequency  are  not  considered. 

Solar  Temperature. — From  the  amount  of  heat  actually 
radiated  by  the  sun,  attempts  have  been  made  to  determine 
the  actual  temperature  of  the  solar  surface.  The  esti- 
mates reached  by  various  authorities  differ  widely,  as  the 
laws  which  govern  the  absorption  within  the  solar  en- 
velope are  almost  unknown.  Some  such  law  of  absorp- 
tion has  to  be  supposed  in  any  such  investigation,  and  the 
estimates  have  differed  widely  according  to  the  adapted 
law. 

SECCHI  estimates  this  temperature  as  about  6,100,000°  C. 
Other  estimates  are  far  lower,  but,  according  to  all  sound 


SPOTS  ON  THE  SUN.  287 

philosophy,  the  temperature  must  far  exceed  any  ter- 
restrial temperature.  There  can  be  no  doubt  that  if  the 
temperature  of  the  earth's  surface  were  suddenly  raised  to 
that  of  the  sun,  no  single  chemical  element  would  remain 
in  its  present  condition.  The  most  refractory  materials 
would  be  at  once  volatilized. 

We  may  concentrate  the  heat  received  upon  several  square  feet 
(the  surface  of  a  huge  burning-lens  or  mirror,  for  instance), 
examine  its  effects  at  the  focus,  and,  making  allowance  for  the  con- 
densation by  the  lens,  see  what  is  the  minimum  possible  tempera- 
ture of  the  sun.  The  temperature  at  the  focus  of  the  lens  cannot 
be  higher  than  that  of  the  source  of  heat  in  the  sun  ;  we  can  only 
concentrate  the  heat  received  on  the  surface  of  the  lens  to  one 
point  and  examine  its  effects.  If  a  lens  three  feet  in  diameter  be 
used,  the  most  refractory  materials,  as  fire-clay,  platinum,  the  dia- 
mond, are  at  once  melted  or  volatilized.  The  effect  of  the  lens  is 
plainly  the  same  as  if  the  earth  were  brought  closer  to  the  sun,  in 
the  ratio  of  the  diameter  of  the  focal  image  to  that  of  the  lens.  In 
the  case  of  the  lens  of  three  feet,  allowing  for  the  absorption,  etc., 
this  distance  is  yet  greater  than  that  of  the  moon  from  the  east, 
so  that  it  appears  that  any  comet  or  planet  so  close  as  this  to  the 
sun,  if  composed  of  materials  similar  to  those  in  the  earth,  must 
be  vaporized. 

If  we  calculate  at  what  rate  the  temperature  of  the  sun  would  be 
lowered  annually  by  the  radiation  from  its  surface,  we  shall  find  it 
to  be  \\°  Centigrade  yearly  if  its  specific  heat  is  that  of  water, 
and  between  3°  and  6°  per  annum  if  its  specific  heat  is  the  same  as 
that  of  the  various  constituents  of  the  earth  itself.  It  would  there- 
fore cool  down  in  a  few  thousand  years  by  an  appreciable  amount. 

§  3.    SUN-SPOTS  AND  FACUL.E. 

A  very  cursory  examination  of  the  sun's  disk  with  a 
small  telescope  will  generally  show  one  or  more  dark  spots 
upon  the  photosphere.  These  are  of  various  sizes,  from 
minute  black  dots  \"  or  2"  in  diameter  (1000  kilometres 
or  less)  to  large  spots  several  minutes  of  arc  in  extent. 

Solar  spots  generally  have  a  dark  central  nucleus  or 
umbra,  surrounded  by  a  border  or  penumbra  of  grayish 
tint,  intermediate  in  shade  between  the  central  blackness 
and  the  bright  photosphere.  By  increasing  the  power  of 
the  telescope,  the  spots  are  seen  to  be  of  very  complex 
forms.  The  umbra  is  often  extremely  irregular  in  shape, 


288 


ASTRONOMY. 


and  is  sometimes  crossed  by  bridges  or  ligaments  of  sinning 
matter.  The  penumbra  *  is  composed  of  filaments  of 
brighter  and  darker  light,  which  are  arranged  in  strige. 
The  appearances  of  the  separate  filaments  are  as  if  they 
were  directed  downward  toward  the  interior  of  the  spot 
in  an  oblique  direction.  The  general  aspect  of  a  spot  un- 
der considerable  magnifying  power  is  shown  in  Fig.  78. 

The  first  printed  account  of  solar  spots  was  given  by 
FABKITIUS  in  1611,  and  GALILEO  in  the  same  year  (May, 
1611)  also  described  them.  They  were  also  attentively 


FlG.    78.  — UMBRA   AND   PENUMBRA   OF    SUN-SPOT. 

studied  by  the  Jesuit  SCHEINER,  who  supposed  them  to  be 
small  planets  projected  against  the  solar  disk.  This  idea 
was  disproved  by  GALILEO,  whose  observations  showed 
them  to  belong  to  the  sun  itself,  a  ad  to  move  uniformly 
across  the  solar  disk  from  east  to  west.  A  spot  just  visible 
at  the  east  limb  of  the  sun  on  any  one  day  travelled  slowly 
across  the  disk  for  12  or  14  days,  when  it  reached  the  west 
limb,  behind  which  it  disappeared.  After  about  the  same 
period,  it  reappeared  at  the  eastern  lirnb,  unless,  as  is  often 
the  case,  it  had  in  the  mean  time  vanished. 


SUN'S  SPOTS  AND  ROTATION.  289 

The  spots  arc  not  permanent  in  their  nature,  but  are 
formed  somewhere  on  the  sun,  and  disappear  after  lasting 
a  few  days,  weeks,  or  months.  But  so  long  as  they  last 
they  move  regularly  from  east  to  west  on  the  sun's  appar- 
ent disk,  making  one  complete  rotation  in  about  25  days. 
This  period  of  25  days  is  therefore  approximately  the  rota- 
tion period  of  the  sun  itself. 

Spotted  Region. — It  is  found  that  the  spots  are  chiefly  con- 
fined to  two  zones,  one  in  each  hemisphere,  extending  from  about 
10°  to  35°  or  40°  of  heliographic  latitude.  In  the  polar  regions, 
spots  are  scarcely  ever  seen,  and  on  the  solar  equator  they  are  much 


FlG.    79. — PHOTOGRAPH   OF   THE    SUN. 

more  rare  than  in  latitudes  10°  north  or  south.  Connected  with 
the  spots,  but  lying  on  or  above  the  solar  surface,  are  faculce,  mot- 
tlings  of  light  brighter  than  the  general  surface  of  the  sun.  The 
formation  of  a  sun-spot  is  said  to  be  often  presaged  by  the  ap- 
pearance of  facula3  near  the  point  where  the  spot  is  to  form. 

Solar  Rotation. — To  obtain  the  exact  period  of  rotation,  the 
spots  must  be  carefully  fixed  in  position  by  micrometric  measures 
from  day  to  day,  the  times  of  the  measures  being  noted.  Better 
still,  daily  photographs  may  be  made  and  afterward  measured. 
This  has  been  done  by  several  observers,  and  the  remarkable  result 
reached  that  the  spots  do  not  all  rotate  exactly  in  the  same  period, 
but  that  this  time,  as  determined  from  any  spot,  depends  upon  the 
heliograpliic  latitude  of  the  spot,  or  its  angular  distance  from  the 


290  ASTRONOMY. 

solar  equator.  A  series  of  observations  made  by  Mr.  CARRINGTON 
of  England  (by  the  eye)  give  the  following  values  of  the  rotation 
times  T,  for  spots  in  different  heliographic  latitudes  L  : 

L  =      0°  5°  10°  15°  20° 

T  =  25 • 187  days      25 • 222  25 • 327  25 • 500  25  - 739 

L  =    25°  30°  35°  40°  45° 

Tr=26-040  26-398  26-804  27-252  27-730 

The  period  of  rotation  seems  also  to  vary  somewhat  in  different 
years  even  for  spots  in  the  same  heliographic  latitude,  so  that  we 
really  cannot  assign  any  one  definite  rotation  time  to  the  sun,  as 
we  can  to  the  earth  or  the  moon. 

"  The  probability  is  that  the  sun,  not  being  solid,  has  really  no  one 
period  of  rotation,  but  different  portions  of  its  surface  and  of  its  in- 
ternal mass  move  at  different  rates,  and  to  some  extent  independent- 
ly of  each  other,  though  approximately  in  one  plane  inclined  about 
7°  to  the  ecliptic,  and  around  a  common  axis.  The  individual 
spots  drift  in  latitude  as  well  as  in  longitude,  and,  on  the  whole,  it 
appears  that  spots  within  15°  or  20°  of  the  solar  equator  on  either 
side  move  toward  the  equator,  while  beyond  this  limit  they  move 
away  from  it."  (YOUNG.) 

Solar  Axis  and  Equator. — The  spots  must  revolve  with  the 
surface  of  the  sun  about  his  axis,  and  the  directions  of  their  motions 
must  be  approximately  parallel  to  his  equator.  Fig.  80  shows 
the  appearances  as  actually  observed,  the  dotted  lines  representing 
the  apparent  paths  of  the  spots  across  the  sun's  disk  at  different 
times  of  the  year.  In  June  and  December  these  paths,  to  an  ob- 
server on  the  earth,  seem  to  be  right  lines,  and  hence  at  these  times 
the  observer  must  be  in  the  plane  of  the  solar  equator.  At  other 
times  the  paths  are  ellipses,  and  in  March  and  September  the 
planes  of  these  ellipses  are  most  oblique,  showing  the  spectator  to 
be  then  furthest  from  the  plane  of  the  solar  equator.  The  incli- 
nation of  the  solar  equator  to  the  ecliptic  is,  as  already  stated,  about 
7°  9',  and  the  axis  of  rotation  is  of  course  perpendicular  to  it. 

Nature  of  the  Spots. — The  sun-spots  are  really  depres- 
sions in  the  photosphere,  as  was  first  pointed  out  by  AN- 
DREW WILSON  of  Glasgow.  When  a  spot  is  seen  at  the 
edge  of  the  disk,  it  appears  as  a  notch  in  the  limb,  and  is 
elliptical  in  shape.  As  the  rotation  carries  it  further  and 
further  on  to  the  disk,  it  becomes  more  and  more  nearly 
circular  in  shape,  and  after  passing  the  centre  of  the  disk 
the  appearances  take  place  in  reverse  order. 

These  observations  were  explained  by  WILSON,  and  more  fully  by 
Sir  WILLIAM  HERSCHEL,  by  supposing  the  sun  to  consist  of  an  in- 
terior dark  cool  mass,  surrounded  by  two  layers  of  clouds.  The 


SOLAR  SPOTS.  291 

outer  layer,  which  forms  the  visible  photosphere,  was  supposed 
extremely  brilliant.  The  inner  layer,  which  could  not  be  seen 
except  when  a  cavity  existed  in  the  photosphere,  was  supposed 
to  be  dark.  The  appearance  of  the  edges  of  a  spot,  which  has 
been  described  as  the  penumbra,  was  supposed  to  arise  from 
those  dark  clouds.  The  spots  themselves  are,  according  to  this 
view,  nothing  but  openings  through  both  of  the  atmospheres,  the 


FlG.    80. — APPARENT  PATH  OF    SOLAR    SPOT  AT  DIFFERENT    SEASONS. 

nucleus  of  the  spot  being  simply  the  black  surface  of  the  inner 
sphere  of  the  sun  itself. 

This  theory,  which  the  figure  on  the  next  page  exemplifies, 
accounts  for  the  facts  as  they  were  known  to  HERSCHEL.  But  when 
it  is  confronted  with  the  questions  of  the  cause  of  the  sun's  heat 
and  of  the  method  by  which  this  heat  has  been  maintained  con- 
stant in  amount  for  centuries,  it  breaks  down  completely.  The 


292 


ASTRONOMY. 


conclusions  of  WILSON  and  HERSJCHEL,  that  the  spots  are  depressions 
in  the  sun's  surface,  are  undoubted.  But  the  existence  of  a  cool  cen- 
tral and  solid  nucleus  to  the  sun  is  now  known  to  be  impossible. 
The  apparently  black  centres  of  the  spots  are  so  mostly  by  contrast. 
If  they  were  seen  against  a  perfectly  black  background,  they  would 
appear  very  bright,  as  has  been  proved  by  the  photometric  measures 
of  Professor  LANGLEY.  And  a  cool  solid  nucleus  beneath  such  an 
atmosphere  as  HERSCHEL  supposed  would  soon  become  gaseous  by 
the  conduction  and  radiation  of  the  heat  of  the  photosphere.  The 
supply  of  solar  heat,  which  has  been  very  nearly  constant  during 
the  historic  period,  would  in  a  sun  so  constituted  have  sensibly 
diminished  in  a  few  hundred  years.  For  these  and  other  reasons, 
the  hypothesis  of  HERSCHEL  must  be  modified,  save  as  to  the  fact 
that  the  spots  are  really  cavities  in  the  photosphere. 


FIG. 


81. — APPEARANCE  OP  A  SPOT  NEAR  THE  LIMB  AND  NEAR  THE 
CENTRE  OP  THE  SUN. 

Number  and  Periodicity  of  Solar  Spots. — The  number 
of  solar  spots  which  come  into  view  varies  from  year  to 
year.  Although  at  first  sight  this  might  seem  to  be  what 
we  call  a  purely  accidental  circumstance,  like  the  occur- 
rence of  cloudy  and  clear  years  on  the  earth,  yet  the  series 
of  observations  of  sun-spots  by  Hofrath  SCHWABE  of 
Dessau  (see  the  table),  continued  by  him  for  forty  years, 
established  the  fact  that  this  number  varied  periodically. 
This  had  indeed  been  previously  suspected  by  HORKEBOW, 


PE1UODIVITY  OF  SUN-SPOTS. 


293 


but  it  was  independently  suggested  and  completely  proved 
by  SCHWABE. 


TABLE  OP  SCHWABE' s  RESULTS. 


YEAR. 

Days  of 
Observation. 

Days  of  no 
Spots. 

New  Groups. 

Mean  Diurnal 
Variation  in 
Declination  of 
the  Magnetic 
Needle. 

1826     

277 

22 

118 

9-75 

1827  

273 

2 

161 

11-33 

1828     .  . 

282 

o 

225 

11-38 

1829... 

244 

0 

199 

14-74 

1830  

217 

1 

190 

12-13 

1831  
1832     . 

239 

270 

3 
49 

149 
84 

12-22 

1833  

247 

139 

33 

1834 

273 

120 

51 

1835     . 

244 

18 

173 

9-57 

183G  •     . 

200 

0 

272 

12-34 

1837  

168 

0 

333 

12-27 

1838 

202 

0 

282 

12-74 

1839 

205 

0 

162 

11-03 

1840  

263 

3 

152 

9-91 

1841  • 

283 

15 

102 

7-82 

1842 

307 

64 

68 

7-08 

1843.    . 

312 

149 

34 

7-15 

1844  

321 

111 

52 

6-61 

1845  

332 

29 

114 

8-13 

1846     . 

314 

1 

157 

8-81 

1847  

276 

0 

257 

9-55 

1848 

278 

o 

330 

11-15 

1849 

285 

o 

238 

10-64 

1850  

308 

2 

186 

10-44 

1851  

308 

0 

151 

8-32 

1852 

337 

2 

125 

'    8-09 

1853  

299 

3 

91 

7-09 

1854  

334 

65 

67 

6-81 

1855 

313 

146 

79 

6-41 

1856     ... 

321 

193 

34 

5-98 

1857  

324 

52 

98 

6-95 

1858  

335 

0 

188 

7-41 

1859. 

343 

o 

205 

10-37 

I860  

332 

0 

211 

10-05 

1861  

322 

0 

204 

9-17 

1862 

317 

3 

160 

8-59 

1863  

330 

2 

124 

8-84 

1864  

325 

4 

130 

8-02 

1865  

307 

25 

93 

8-14 

1866  .  .  . 

349 

76 

45 

7-65 

1867  . 

316 

195 

25 

7-09 

1868 

301 

23 

101 

8-15 

294  ASTRONOMY. 

The  periodicity  of  the  spots  is  evident  from  the  table. 
It  will  appear  in  a  more  striking  way  from  the  following 
summary  : 

From  1828  to  1831,  sun  without  spots  on  only 1  day. 

In  1833,  "  "  "  ....  139  days. 

From  1836  to  1840,  "  "  "            3      " 

In  1843,  "  "  "  ....  147      " 

From  1847  to  1851,  "  "  "  ....        2      " 

In  1856,  "  "  "  ....  193      " 

From  1858  to  1861,  "  "  "            no  day. 

In  1867,  "  "                          195  days. 

Every  11  years  there  is  a  minimum  number  of  spots, 
and  about  5  years  after  each  minimum  there  is  a  maxi- 
mum. If  instead  of  merely  counting  the  number  of  spots, 
measurements  are  made  on  solar  photograms,  as  they 
are  called,  of  the  extent  of  spotted  area,  the  period  comes 
out  with  greater  distinctness.  This  periodicity  of  the 
area  of  the  solar  spots  appears  to  be  connected  with  mag- 
netic phenomena  on  the  earth's  surface,  and  with  the  num- 
ber of  auroras  visible.  It  has  been  supposed  to  be  con- 
nected also  with  variations  of  temperature,  of  rainfall, 
and  with  other  meteorological  phenomena  such  as  the  mon- 
soons of  the  Indian  Ocean,  etc.  The  cause  of  this  period- 
icity is  as  yet  unknown.  CARRINGTON,  DE  LA  HUE, 
LOEWY,  and  STEWART  have  given  reasons  which  go  to  show 
that  there  is  a  connection  between  the  spotted  area  and  the 
configurations  of  the  planets,  particularly  of  Jupiter, 
Venus,  and  Mercury.  ZOLLNER  says  that  the  cause  lies 
within  the  sun  itself,  and  assimilates  it  to  the  periodic 
action  of  a  geyser,  which  seems  to  be  h  priori  probable. 
Since,  however,  the  periodic  variations  of  the  spots  cor- 
respond to  the  magnetic  variation,  as  exhibited  in  the  last 
column  of  the  table  of  SCHWABE'S  results,  it  appears  that 
there  may  be  some  connection  of  an  unknown  nature 
between  the  sun  and  the  earth  at  least.  But  at  present 
we  can  only  state  our  limited  knowledge  and  wait  for 
further  information. 


PERIODICITY  OF  SUN-SPOTS. 


295 


Dr.  WOLF  (Director  of  the  Zurich  Observatory)  has  col- 
lected all  the  available  observations  of  the  solar  spots,  and 
it  is  found  that  since  1610  we  have  a  tolerably  complete 
record  of  these  appearances.  The  number  and  character 
of  the  spots  are  now  noted  every  day  by  observers  in  many 
quarters  of  the  civilized  world.  This  long  series  of  obser- 
vations has  served  as  a  basis  to  determine  each  epoch  of 
maximum  and  minimum  which  has  occurred  since  1610, 
and  from  thence  to  determine  the  length  of  each  single 
period. 

The  following  table  gives  Dr.  WOLF'S  results  : 

TABLE  GIVING  THE  TIMES  OF  MAXIMUM  AND  MINIMUM  SUN-SPOT 
FREQUENCY,  ACCORDING  TO  WOLF. 


FIRST  SERIES. 

SECOND  SERIES. 

Minima. 

Diff. 

Maxima. 

Diff. 

Minima. 

Diff. 

Maxima. 

Diff. 

A.D.  1610-8 

1615-5 

1745-0 

1750-3 

8-2 

10-5 

10-2 

11-2 

1619-0 

1626-0 

1755-2 

1761-5 

15-0 

13-5 

11-3 

8-2 

1634-0 

1639-5 

1766-5 

1769-7 

11-0 

9-5 

9-0 

8-7- 

1645-0 

1649-0 

1775-5 

1778-4 

10-0 

11-0 

9-2 

9-7 

1655-0 

1660-0 

1784-7 

1788-1 

11-0 

15-0 

13-6 

16-1 

1666-0 

1675-0 

1798-3 

1804-2 

13-5 

10-0 

12-3 

12-2 

1679-5 

1685-0 

1810-6 

1816-4 

10-0 

8-0 

12-7 

13-5 

1689-5 

1693-0 

1823-3 

1829-9 

8-5 

12-5 

10-6 

7-3 

1698-0 

1705-5 

1833-9 

1837-2 

14-0 

12-7 

9-6 

10-9 

1712-0 

1718-2 

1843-5 

1848-1 

11-5 

9-3 

12-5 

12-0 

1723-5 

1727-5 

1856-0 

1860-1 

10-5 

11-2 

11-2 

10-5 

1734-0 

1738-7 

1867-2 

1870  -I 

11  -20  ±2  -11  years. 

ll-20±2-06ys. 

ll-ll±l-54ys. 

10-94±2-52ys. 

±0-64 

±0-63 

±0-47 

±0-76 

296  ASTRONOMY. 

From  the  first  series  of  earlier  observations,  the  period 
comes  out  from  observed  viinima,  11-20  years,  with  a 
variation  of  two  years  ;  from  observed  maxima  the  period 
is  11-20  years,  with  variation  of  three  years  —  that  is,  this 
series  shows  the  period  to  vary  between  13-3  and  9-1 
years.  If  we  suppose  these  errors  to  arise  only  from  errors 
of  observation,  and  not  to  be  real  changes  of  the  period 
itself,  the  mean  period  is  11-20  ±  0-64. 

The  results  from  the  second  series  are  also  given  at 
the  foot  of  the  table.  From  a  combination  of  the  two,  it 
follows  that  the  mean  period  is  11-111  ±  0-307  years, 
with  an  oscillation  of  ±  2  •  030  years. 

These  results  are  formulated  by  Dr.  WOLF  as  follows  : 
The  frequency  of  solar  spots  has  continued  to  change 
periodically  since  their  discovery  in  1610  ;  the  mean  length 
of  the  period  is  11^  years,  and  the  separate  periods  may 
differ  from  this  mean  period  by  as  much  as  2-03  years. 

A  general  relation  between  the  frequency  of  the  spots  and  the 
variation  of  the  magnetic  needle  is  shown  by  the  numbers  which 
have  been  given  in  the  table  of  SCHWABE'S  results.  This  relation 
has  been  most  closely  studied  by  WOLF.  He  denotes  by  g  the 
number  of  groups  of  spots  seen  on  any  day  on  the  sun,  counting 
each  isolated  spot  as  a  group  ;  by  /is  denoted  the  number  of  spots 
in  each  group  (fg  is  then  proportional  to  the  spotted  area)  ;  by  Tc  a 
coefficient  depending  upon  the  size  of  the  telescope  used  for  obser- 
vation, and  by  r  the  daily  relative  number  so  called  ;  then  he  sup. 
poses 


From  the  daily  relative  numbers  are  formed  the  mean  monthly 
and  the  mean  annual  relative  numbers  r.  Then,  according  to 
WOLF,  if  ®  is  the  mean  annual  variation  of  the  magnetic  needle  at 
any  place,  two  constants  for  that  place,  a  and  /?,  can  be  found,  so 
that  the  following  formula  is  true  for  all  years  : 

t>  =  a  +  j3  -  r. 
Thus  for  Munich  the  formula  becomes, 

v  =  6'-27  +  0'-051  r; 
and  for  Prague, 

0  =  5'  -80  +  0'-045  r,  and  so  on. 


TOTAL  ECLIPSES  OF  THE  SUN. 


297 


YFATI 

MUNICH. 

PKAQUB. 

Observed. 

Computed. 

A 

Observed. 

Computed. 

A 

1870.. 
1871  

12-27 
11-70 

12-77 
11-56 

-0-50 
+  0-14 

11-41 
11-60 

12-10 
10-89 

-0-69 
+  0-71 

1872  
1873  

10-96 
9-12 

11-13 
9-54 

-0-17 
—  0-42 

10-70 
9-05 

10-46 

8-87 

+  0-24 
+  0-18 

The  above  comparison  bears  out  the  conclusion  that  the 
magnetic  variations  are  subjected  to  the  same  perturba- 
tions as  the  development  of  the  solar  spots,  and  it  may 
be  said  that  the  changes  in  the  frequency  of  solar  spots 
and  the  like  changes  of  magnetic  variations  show  that 
these  two  phenomena  are  dependent  the  one  on  the  other, 
or  rather  upon  the  same  cosmical  cause.  What  this  cause 
is  remains  as  yet  unknown. 

§  4.    THE  SUN'S  CHROMOSPHERE  AND  CORONA. 

Phenomena  of  Total  Eclipses. — The  beginning  of  a 
total  solar  eclipse  is  an  insignificant  phenomenon.  It  is 
marked  simply  by  the  small  black  notch  made  in  the  lu- 
minous disk  of  the  sun  by  the  advancing  edge  or  limb  of 
the  moon.  This  always  occurs  on  the  western  half  of  the 
sun,  as  the  moon  moves  from  west  to  east  in  its  orbit.  An 
hour  or  more  must  elapse  before  the  moon  has  advanced 
sufficiently  far  in  its  orbit  to  cover  the  sun's  disk.  During 
this  time  the  disk  of  the  sun  is  gradually  hidden  until  it 
becomes  a  thin  crescent.  To  the  general  spectator  there 
is  little  to  notice  during  the  first  two  thirds  of  this  period 
from  the  beginning  of  the  eclipse,  unless  it  be  perhaps  the 
altered  shapes  of  the  images  formed  by  small  holes  or 
apertures.  Under  ordinary  circumstances,  the  image  of 
the  sun,  made  by  the  solar  rays  which  pass  through  a  small 
hole— in  a  card,  for  example— are  circular  in  shape,  like  the 
shape  of  the  sun  itself.  When  the  sun  is  crescent,  the 


298  ASTRONOMY. 

image  of  the  sun  formed  by  such  rajs  is  also  crescent, 
and,  under  favorable  circumstances,  as  in  a  thick  forest 
where  the  interstices  of  the  leaves  allow  such  images  to  be 
formed,  the  effect  is  quite  striking.  The  reason  for  this 
phenomenon  is  obvious. 

The  actual  amount  of  the  sun's  light  may  be  diminished 
to  two  thirds  or  three  fourths  of  its  ordinary  amount  with- 
out its  being  strikingly  perceptible  to  the  eye.  What  is 
first  noticed  is  the  change  which  takes  place  in  the  color 
of  the  surrounding  landscape,  which  begins  to  wear  a  rud- 
dy aspect.  This  grows  more  and  more  pronounced,  and 
gives  to  the  adjacent  country  that  weird  effect  which  lends 
so  much  to  the  impressiveness  of  a  total  eclipse.  The  rea- 
son for  the  change  of  color  is  simple.  We  have  already 
said  that  the  sun's  atmosphere  absorbs  a  large  proportion 
of  the  bluer  rays,  and  as  this  absorption  is  dependent  on 
the  thickness  of  the  solar  atmosphere  through  which  the 
rays  must  pass,  it  is  plain  that  just  before  the  sun  is  total- 
ly covered  the  rays  by  which  we  see  it  will  be  redder  than 
ordinary  sunlight,  as  they  are  those  which  come  from 
points  near  the  sun's  limb,  where  they  have  to  pass  through 
the  greatest  thickness  of  the  sun's  atmosphere. 

The  color  of  the  light  becomes  more  and  more  lurid  up 
to  the  moment  when  the  sun  has  nearly  disappeared.  If 
the  spectator  is  upon  the  top  of  a  high  mountain,  he  can 
then  begin  to  see  the  moon's  shadow  rushing  toward  him 
at  the  rate  of  a  mile  in  about  two  seconds.  Just  as  the 
shadow  reaches  him  there  is  a  sudden  increase  of  darkness 
— the  brighter  stars  begin  to  shine  in  the  dark  lurid  sky, 
the  thin  crescent  of  the  sun  breaks  up  into  small  points  or 
dots  of  light,  which  suddenly  disappear,  and  the  moon  it- 
self, an  intensely  black  ball,  appears  to  hang  isolated  in  the 
heavens. 

An  instant  afterward,  the  corona  is  seen  surrounding  the 
black  disk  of  the  moon  with  a  soft  effulgence  quite  differ- 
ent from  any  other  light  known  to  us.  Near  the  moon's 
limb  it  is  intensely  bright,  and  to  the  naked  eye  uniform 


TOTAL  ECLIPSES  OF  THE  SUN.  299 

in  structure  ;  5'  or  10'  from  the  limb  this  inner  corona 
has  a  boundary  more  or  less  defined,  and  from  this  extend 
streamers  and  wings  of  fainter  and  more  nebulous  light. 
These  are  of  various  shapes,  sizes,  and  brilliancy.  No 
two  solar  eclipses  yet  observed  have  been  alike  in  this  re- 
spect. 

These  wings  seem  to  vary  from  time  to  time,  though  at 
nearly  every  eclipse  the  same  phenomena  are  described  by 
observers  situated  at  different  points  along  the  line  of 
totality.  That  is,  these  appearances,  though  changeable, 
do  not  change  in  the  time  the  moon's  shadow  requires  to 
pass  from  Vancouver's  Island  to  Texas,  for  example,  which 
is  some  fifty  minutes. 

Superposed  upon  these  wings  may  be  seen  (sometimes 
with  the  naked  eye)  the  red  flames  or  protuberances  which 
were  first  discovered  during  a  solar  eclipse.  These  need 
not  be  more  closely  described  here,  as  they  can  now  be 
studied  at  any  time  by  aid  of  the  spectroscope. 

The  total  phase  lasts  for  a  few  minutes  (never  more  than 
six  or  seven),  and  during  this  time,  as  the  eye  becomes  more 
and  more  accustomed  to  the  faint  light,  the  outer  corona  is 
seen  to  stretch  further  and  further  away  from  the  sun's 
limb.  At  the  eclipse  of  1878,  July  29th,  it  was  seen  by 
Professor  LANGLEY,  and  by  one  of  the  writers,  to  extend 
more  than  6°  (about  9,000,000  miles)  from  the  sun's  limb. 
Just  before  the  end  of  the  total  phase  there  is  a  sudden 
increase  of  the  brightness  of  the  sky,  due  to  the  increased 
illumination  of  the  earth's  atmosphere  near  the  observer, 
and  in  a  moment  more  the  sun's  rays  are  again  visible, 
seemingly  as  bright  as  ever.  From  the  end  of  totality  till 
the  last  contact  the  phenomena  of  the  first  half  of  the 
eclipse  are  repeated  in  inverse  order. 

Telescopic  Aspect  of  the  Corona. —  Such  are  the  ap- 
pearances to  the  naked  eye.  The  corona,  as  seen  through 
a  telescope,  is,  however,  of  a  very  complicated  structure. 
The  inner  corona  is  usually  composed  of  bright  striae  or  fil- 
aments separated  by  darker  bands,  and  some  of  these  lat- 


300  ASTRONOMY. 

ter  are  sometimes  seen  to  be  almost  totally  black.  The 
appearances  are  extremely  irregular,  but  they  are  often  as 
if  the  inner  corona  were  made  up  of  brushes  of  light  on  a 
darker  background.  The  direction  of  these  brushes  is 
often  radial  to  the  sun,  especially  about  the  poles,  but 
where  the  outer  corona  joins  on  to  the  inner  these  brushes 
are  sometimes  bent  over  so  as  to  join,  as  it  were,  the 
boundaries  of  the  outer  light. 

The  great  difficulties  in  the  way  of  studying  the  corona 
have  been  due  to  the  short  time  at  the  disposal  of  the  ob- 
server, and  to  the  great  differences  which  even  the  best 
draughtsmen  will  make  in  their  rapid  sketches  of  so  com- 
plicated a  phenomenon.  The  figure  of  the  inner  corona 
on  the  next  page  is  a  copy  of  one  of  the  best  drawings  made 
of  the  eclipse  of  1869,  and  is  inserted  chiefly  to  show  the 
nature  of  the  only  drawings  possible  in  the  limited  time. 
The  numbers  refer  to  the  red  prominences  around  the  limb . 
The  radial  structure  of  the  corona  and  its  different  exten- 
sion and  nature  at  different  points  are  also  indicated  in  the 
drawing. 

The  figure  on  page  302,  is  acopy  of  a  crayon  drawing  made  in  1878. 
The  best  evidence  which  we  can  gain  of  the  details  of  the  corona 
comes,  however,  from  a  series  of  photographs  taken  during  the  whole 
of  totality.  A  photograph  with  a  short  exposure  gives  the  details 
of  the  inner  corona  well,  but  is  not  affected  by  the  fainter  outlying 
parts.  One  of  longer  exposure  shows  details  further  away  from 
the  sun's  limb,  while  those  near  it  are  lost  in  a  glare  of  light,  being 
over-exposed,  and  so  on.  In  this  way  a  series  of  photographs 
gives  us  the  means  of  building  up,  as  it  were,  the  whole  corona 
from  its  brightest  parts  near  the  sun's  limb  out  to  the  faintest  por- 
tions which  will  impress  themselves  on  a  photographic  plate. 

The  corona  and  red  prominences  are  solar  appendages. 
It  was  formerly  doubtful  whether  the  corona  was  an 
atmosphere  belonging  to  the  sun  or  to  the  moon.  At  the 
eclipse  of  1860  it  was  proved  by  measurements  that  the 
red  prominences  belonged  to  the  sun  and  not  to  the  moon, 
since  the  moon  gradually  covered  them  by  its  motion, 
they  remaining  attached  to  the  sun.  The  corona  has  also 
since  been  shown  to  be  a  solar  appendage. 


TOTAL  ECLIPSES  OF  THE  SUN. 


301 


The  eclipse  of  1851  was  total  in  Sweden  and  neigh- 
boring parts,  and  was  very  carefully  observed.  Similar 
prominences  were  seen  about  the  sun's  limb,  and  one  of 
so  bizarre  a  form  as  to  show  that  it  could  by  no  possibility 


FlQ.  82. — DRAWING  OF  THE  CORONA  MADE  DURING  THE  ECLIPSE  OF 
AUGUST  7,  1869. 

be  a  mountain  or  solid  mass,  since  if  such  had  been  the 
case  it  would  inevitably  have  overturned.  It  was  there- 
fore a  gaseous  or  cloud -like  appendage  belonging  to  the 


302 


ASTRONOMY. 


FIG.  83. — SUN'S  COKONA  DURING  THE  ECLIPSE  OF  JULY  29,  1878. 


THE  SUN'S  PROMINENCES.  303 

sun.  There  were  others  of  various  and  perhaps  varying 
shapes,  and  the  bases  of  these  were  connected  by  a  low 
band  of  serrated  rose-colored  light.  One  of  these  protu- 
berances was  shown  to  be  entirely  above  the  sun,  as  if 
floating  within  its  atmosphere.  Around  the  whole  disk 
of  the  sun  a  ring  of  similar  nature  to  the  prominences 
exists,  which  is  brighter  than  the  corona,  and  seems  to 
form  a  base  for  the  protuberances  themselves  ;  this  is 
the  sierra.  Some  of  the  red  flames  were  of  enormous 
height ;  one  of  at  least  80,000  miles. 


.    84. — FORMS  OP  THE   SOLAR  PROMINENCES   AS   SEEN  WITH  THE 
SPECTROSCOPE. 

Gaseous  Nature  of  the  Prominences. — The  next  eclipse 
(1868,  July)  was  total  in  India,  and  was  observed  by  many 
skilled  astronomers.  A  discovery  of  M.  JANSSEN'S*  will 
make  this  eclipse  forever  memorable.  lie  was  provided 
with  a  spectroscope,  and  by  it  observed  the  prominences. 
One  prominence  in  particular  was  of  vast  size,  and  when 
the  spectroscope  was  turned  upon  it,  its  spectrum  was  dis- 
continuous, showing  the  bright  lines  of  hydrogen  gas. 

*  Now  Director  of  the  Solar  Observatory  of  Mention,  near  Paris. 


304  ASTRONOMY. 

The  brightness  of  the  spectrum  was  so  marked  that 
JANSSEN  determined  to  keep  his  spectroscope  fixed  upon  it 
even  after  the  reappearance  of  sunlight,  to  see  how  long  it 
could  be  followed.  It  was  found  that  its  spectrum  could 
still  be  seen  after  the  return  of  complete  sunlight ;  and  not 
only  on  that  day,  but  on  subsequent  days,  similar  phenom- 
ena could  be  observed. 

One  great  difficulty  was  conquered  in  an  instant.  The 
red  flames  which  formerly  were  only  to  be  seen  for  a  few 
moments  during  the  comparatively  rare  occurrences  of 
total  eclipses,  and  whose  observation  demanded  long  and 
expensive  journeys  to  distant  parts  of  the  world,  could 
now  be  regularly  observed  with  all  the  facilities  offered  by 
a  fixed  observatory. 

This  great  step  in  advance  was  independently  made  by 
Mr.  LOCKYER,*  and  his  discovery  was  derived  from  pure 
theory,  unaided  by  the  eclipse  itself.  By  this  method 
the  prominences  have  been  carefully  mapped  day  by 
day  all  around  the  sun,  and  it  has  been  proved  that 
around  this  body  there  is  a  vast  atmosphere  of  hydrogen 
gas — the  chromosphere  or  sierra.  From  out  of  this  the 
prominences  are  projected  sometimes  to  heights  of  100,000 
kilometres  or  more. 

It  will  be  necessary  to  recall  the  main  facts  of  observation  which  are 
fundamental  in  the  use  of  the  spectroscope.  When  a  brilliant  point  is 
examined  with  the  spectroscope,  it  is  spread  out  by  the  prism  into  a 
band — the  spectrum.  Using  two  prisms,  the  spectrum  becomes  longer, 
but  the  light  of  the  surface,  being  spread  over  a  greater  area,  is  en- 
feebled. Three,  four,  or  more  prisms  spread  out  the  spectrum  propor- 
tionally more.  If  the  spectrum  is  of  an  incandescent  solid  or  liquid,  it 
is  always  continuous,  and  it  can  be  enfeebled  to  any  degree ;  so  that 
any  part  of  it  can  be  made  as  feeble  as  desired. 

This  method  is  precisely  similar  in  principle  to  the  use  of  the  telescope 
in  viewing  stars  in  the  daytime.  The  telescope  lessens  the  brilliancy 
of  the  sky,  while  the  disk  of  the  star  is  kept  of  the  same  intensity, 
as  it  is  a  point  in  itself.  It  thus  becomes  visible.  If  it  is  a  glowing  gas, 
its  spectrum  will  consist  of  a  definite  number  of  lines,  say  three— A,  13, 
C,  for  example.  Now  suppose  the  spectrum  of  this  gas  to  be  superposed 
on  the  continuous  spectrum  of  the  sun  ;  by  using  only  one  prism,  the 

*  Mr.  J.  NORMAN  LOCKYER,  F.R.S.,  of  London,  now  attached  to 
the  Science  and  Art  Department  of  the  South  Kensington  Museum. 


THE  SUN'S  HEAT.  305 

solar  spectrum  is  short  and  brilliant,  and  every  part  of  it  may  be  more 
brilliant  than  the  line  spectrum  of  the  gas.  By  increasing  the  disper- 
sion (the  number  of  prisms),  the  solar  spectrum  is  proportionately  en- 
feebled. If  the  ratio  of  the  light  of  the  bodies  themselves,  the  sun  and 
the  gas,  is  not  too  great,  the  continuous  spectrum  may  be  so  enfeebled 
that"  the  line  spectrum  will  be  visible  when  superposed  upon  it,  and 
the  spectrum  of  the  gas  may  then  be  seen  even  in  the  presence  of  true 
sunlight.  Such  "was  the  process  imagined  and  successfully  carried  out 
by  Mr.  LOCKYER,  and  such  is  in  essence  the  method  of  viewing  the 
prominences  to-day  adopted. 

The  Coronal  Spectrum.— In  1869  (August  7th)  a  total  solar 
eclipse  was  visible  in  the  United  States.  It  was  probably  observed 
by  more  astronomers  than  any  preceding  eclipse.  Two  American 
astronomers,  Professor  YOUNG,  of  Dartmouth  College,  and  Professor 
HARKNESS,  of  the  Naval  Observatory,  especially  observed  the  spec- 
trum of  the  corona.  This  spectrum  was  found  to  consist  of  one 
faint  greenish  line  crossing  a  faint  continuous  spectrum.  The 
place  of  this  line  in  the  maps  of  the  solar  spectrum  published  by 
KiiiCHHOFP  was  occupied  by  a  line  which  he  had  attributed  to  the 
iron  spectrum,  and  which  had  been  numbered  1474  in  his  list,  so 
that  it  is  now  spoken  of  as  1474  K.  This  line  is  probably  due  to 
some  gas-  which  must  be  present  in  large  and  possibly  variable 
quantities  in  the  corona,  and  which  is  not  known  to  us  on  the  earth, 
in  this  form  at  least.  It  is  probably  a  gas  even  lighter  than  hydro- 
gen, as  the  existence  of  this  line  has  been  traced  10'  or  20'  from 
the  sun's  limb  nearly  all  around  the  disk. 

In  the  eclipse  of  July  29th,  1878,  which  was  total  in  Colorado 
and  Texas,  the  continuous  spectrum  of  the  corona  was  found  to  be 
crossed  by  the  dark  lines  of  the  solar  spectrum,  showing  that  the 
coronal  light  was  composed  in  part  of  reflected  sunlight. 


§   5.    SOURCES  OP  THE  SUN'S  HEAT. 

Theories  of  the  Sun's  Constitution.  —  No  considerable 
fraction  of  the  heat  radiated  from  the  sun  returns  to  it 
from  the  celestial  spaces,  since  if  it  did  the  earth  would 
intercept  some  of  the  returning  rays,  and  the  temperature 
of  night  would  be  more  like  that  of  noonday.  But  we 
know  the  sun  is  daily  radiating  into  space  2,170,000,000 
times  as  much  heat  as  is  daily  received  by  the  earth,  and 
it  follows  that  unless  the  supply  of  heat  is  infinite  (which 
we  cannot  believe),  this  enormous  daily  radiation  must  in 
time  exhaust  the  supply.  When  the  supply  is  exhausted, 
or  even  seriously  trenched  upon,  the  result  to  the  inhab- 
itants of  the  earth  will  be  fatal.  A  slow  diminution  of 


306  ASTRONOMY. 

the  daily  supply  of  heat  would  produce  a  slow  change  of 
climates  from  hotter  toward  colder.  The  serious  results 
of  a  fall  of  50°  in  the  mean  annual  temperature  of  the 
earth  will  be  evident  when  we  remember  that  such  a  fall 
would  change  the  climate  of  France  to  that  of  Spitzber- 
gen.  The  temperature  of  the  sun  cannot  be  kept  up  by 
the  mere  combustion  of  its  materials.  If  the  sun  were 
solid  carbon,  and  if  a  constant  and  adequate  supply  of 
oxygen  were  also  present,  it  has  been  shown  that,  at  the 
present  rate  of  radiation,  the  heat  arising  from  the  com- 
bustion of  the  mass  would  not  last  more  than  5000  years. 

An  explanation  of  the  solar  heat  and  light  has  been 
suggested,  which  depends  upon  the  fact  that  great  amounts 
of  heat  and  light  are  produced  by  the  collision  of  two 
rapidly  moving  heavy  bodies,  or  even  by  the  passage  of 
a  heavy  body  like  a  meteorite  through  the  earth's  atmos- 
phere. In  fact,  if  we  had  a  certain  mass  available  with 
which  to  produce  heat  in  the  sun,  and  if  this  mass  were  of 
the  best  possible  materials  to  produce  heat  by  burning, 
it  can  be  shown  that,  by  burning  it  at  the  surface  of  the 
sun,  we  should  produce  vastly  less  heat  than  if  we  simply 
allowed  it  to  fall  into  the  sun.  In  the  last  case,  if  it  fell 
from  the  earth's  distance,  it  would  give  6000  times  more 
heat  than  by  its  burning. 

The  least  velocity  with  which  a  body  from  space  could 
fall  upon  the  sun's  surface  is  in  the  neighborhood  of  280 
miles  in  a  second  of  time,  and  the  velocity  may  be  as  great 
as  350  miles.  From  these  facts,  the  meteoric  theory  of 
solar  heat  originated.  It  is  in  effect  that  the  heat  of  the 
sun  is  kept  up  by  the  impact  of  meteors  upon  its  surface. 

No  doubt  immense  numbers  of  meteorites  fall  into  the 
sun  daily  and  hourly,  and  to  each  one  of  them  a  certain 
considerable  portion  of  heat  is  due.  It  is  found  that,  to 
account  for  the  present  amount  of  radiation,  meteorites 
equal  in  mass  to  the  whole  earth  would  have  to  fall  into 
the  sun  every  century.  It  is  extremely  improbable  that  a 
mass  one  tenth  as  large  as  this  is  added  to  the  sun  in  this 


SUPPLY  OF  SOLAR  HEAT.  307 

way  per  century,  if  for  no  other  reason  because  the  earth 
itself  and  every  planet  would  receive  far  more  than  its 
present  share  of  meteorites,  and  would  itself  become  quite 
hot  from  this  cause  alone. 

There  is  still  another  way  of  accounting  for  the  sun's 
constant  supply  of  energy,  and  this  has  the  advantage  of 
appealing  to  no  cause  outside  of  the  sun  itself  in  the  ex- 
planation. It  is  by  supposing  the  heat,  light,  etc. ,  to  be 
generated  by  a  constant  and  gradual  contraction  of  the 
dimensions  of  the  solar  sphere.  As  the  globe  cools  by 
radiation  into  space,  it  must  contract.  In  so  contracting  its 
ultimate  constituent  parts  are  drawn  nearer  together  by 
their  mutual  attraction,  whereby  a  form  of  energy  is  de- 
veloped which  can  be  transformed  into  heat,  light,  elec- 
tricity, or  other  physical  forces. 

This  theory  is  in  complete  agreement  with  the  known 
laws  of  force.  It  also  admits  of  precise  comparison  with 
facts,  since  the  laws  of  heat  enable  us,  from  the  known 
amount  of  heat  radiated,  to  infer  the  exact  amount  of  con- 
traction in  inches  which  the  linear  dimensions  of  the  sun 
must  undergo  in  order  that  this  supply  of  heat  may  be 
kept  unchanged,  as  it  is  practically  found  to  be.  With 
the  present  size  of  the  sun,  it  is  found  that  it  is  only 
necessary  to  suppose  that  its  diameter  is  diminishing  at  the 
rate  of  about  220  feet  per  year,  or  4  miles  per  century, 
in  order  that  the  supply  of  heat  radiated  shall  be  constant. 
It  is  plain  that  such  a  change  as  this  may  be  taking  place, 
since  we  possess  no  instruments  sufficiently  delicate  to 
have  detected  a  change  of  even  ten  times  this  amount 
since  the  invention  of  the  telescope. 

It  may  seem  a  paradoxical  conclusion  that  the  cooling 
of  a  body  may  cause  it  to  become  hotter.  This  indeed  is 
true  only  when  we  suppose  the  interior  to  be  gaseous,  and 
not  solid  or  liquid.  It  is,  however,  proved  by  theory  that 
this  law  holds  for  gaseous  masses. 

If  a  spherical  mass  of  gas  be  condensed  to  one  half  the  primitive 
diameter,  the  central  attraction  upon  any  part  of  its  mass  will  be  in- 


308  ASTRONOMY. 

creased  fourfold,  while  the  surface  subjected  to  this  attraction  will 
be  reduced  to  one  fourth.  Hence  the  pressure  per  unit  of  surface 
will  be  augmented  sixteen  times,  while  the  density  will  be  increased 
but  eight  times.  If  the  elastic  and  the  gravitating  forces  were  in 
equilibrium  in  the  original  condition  of  the  mass,  the  temperature 
must  be  doubled  in  order  that  they  may  still  be  in  equilibrium  when 
the  diameter  is  reduced  to  one  half. 

If,  however,  the  primitive  body  is  originally  solid  or  liquid,  or  if, 
in  the  course  of  time,  it  becomes  so,  then  this  law  ceases  to  hold,  and 
radiation  of  heat  produces  a  lowering  of  the  temperature  of  the 
body,  which  progressively  continues  until  it  is  finally  reduced  to  the 
temperature  of  surrounding  space. 

We  cannot  say  whether  the  sun  has  yet  begun  to  liquefy 
in  his  interior  parts,  and  hence  it  is  impossible  to  predict 
at  present  the  duration  of  his  constant  radiation.  Theory 
shows  us  that  after  about  5, 000, 000  years,  the  sun  radiating 
heat  as  at  present,  and  still  remaining  gaseous,  will  be  re- 
duced to  one  half  of  its  present  volume.  It  seems  prob- 
able that  somewhere  about  this  time  the  solidification 
will  have  begun,  and  it  is  roughly  estimated,  from  this 
line  of  argument,  that  the  present  conditions  of  heat  radi- 
ation cannot  last  greatly  over  10,000,000  years. 

The  future  of  the  sun  (and  hence  of  the  earth)  cannot, 
as  we  see,  be  traced  with  great  exactitude.  The  past  can 
be  more  closely  followed  if  we  assume  (which  is  tolerably 
safe)  that  the  sun  up  to  the  present  has  been  a  gaseous,  and 
not  a  solid  or  liquid  mass.  Four  hundred  years  ago, 
then,  the  sun  was  about  100  miles  greater  in  diameter 
than  now  ;  and  if  we  suppose  this  process  of  contrac- 
tion to  have  regularly  gone  on  at  the  same  rate  (an 
uncertain  supposition),  we  can  fix  a  date  when  the  sun 
filled  any  given  space,  out  even  to  the  orbit  of  Nep- 
tune— that  is,  to  the  time  when  the  solar  system  consisted 
of  but  one  body,  and  that  a  gaseous  or  nebulous  one. 
It  will  subsequently  be  seen  that  the  ideas  here  reached 
a  posteriori  have  a  striking  analogy  to  the  a  priori  ideas 
of  KANT  and  LA  PLACE. 

It  is  not  to  be  taken  for  granted,  however,  that  the 
amount  of  heat  to  be  derived  from  the  contraction  of  the 


AGE  OF  THE  SUN.  309 

sun's  dimensions  is  infinite,  no  matter  how  large  the  prim- 
itive dimensions  may  have  been.  A  body  falling  from 
any  distance  to  the  sun  can  only  have  a  certain  finite  veloc- 
ity depending  on  this  distance  and  the  mass  of  the  sun 
itself,  which,  even  if  the  fall  be  from  an  infinite  distance, 
cannot  exceed,  for  the  sun,  350  miles  per  second.  In 
the  same  way  the  amount  of  heat  generated  by  the  con- 
traction of  the  sun's  volume  from  any  size  to  any  other  is 
finite,  and  not  infinite. 

It  lias  been  shown  that  if  the  sun  has  always  been 
radiating  heat  at  its  present  rate,  and  if  it  had  originally 
filled  all  space,  it  has  required  18, 000, 000  years  to  contract 
to  its  present  volume.  In  other  words,  assuming  the  pres- 
ent rate  of  radiation,  and  taking  the  most  favorable  case, 
the  age  of  the  sun  does  not  exceed  18,000,000  years.  The 
earth,  is  of  course,  less  aged.  The  supposition  lying  at  the 
base  of  this  estimate  is  that  the  radiation  of  the  sun  has 
been  constant  throughout  the  whole  period.  This  is  quite 
unlikely,  and  any  changes  in  this  datum  affect  greatly  the 
final  number  of  years  which  we  have  assigned.  "While 
this  number  may  be  greatly  in  error,  yet  the  method  of 
obtaining  it  seems,  in  the  present  state  of  science,  to  be 
satisfactory,  and  the  main  conclusion  remains  that  the  past 
of  the  sun  is  finite,  and  that  in  all  probability  its  future  is 
a  limited  one.  The  exact  number  of  centuries  that  it  is  to 
last  are  of  no  moment  even  were  the  data  at  hand  to  ob- 
tain them  :  the  essential  point  is,  that,  so  far  as  we  can 
see,  the  sun,  and  incidentally  the  solar  system,  has  a  finite 
past  and  a  limited  future,  and  that^  like  other  natural  ob- 
jects, it  passes  through  its  regular  stages  of  birth,  vigor, 
decay,  and  death,  in  one  order  of  progress. 


CHAPTER    III. 

THE   INFERIOR   PLANETS. 

§    1.    MOTIONS  AND  ASPECTS. 

THE  inferior  planets  are  those  whose  orbits  lie  between 
the  sun  and  the  orbit  of  the  earth.  Commencing  with  the 
more  distant  ones,  they  comprise  Venus,  Mercury ',  and,  in 
the  opinion  of  some  astronomers,  a  planet  called  Vulcan, 
or  a  group  of  planets,  inside  the  orbit  of  Mercury.  The 
planets  Mercury  and  Venus  have  so  much  in  common  that 
a  large  part  of  what  we  have  to  say  of  one  can  be  applied 
to  the  other  with  but  little  modification. 

The  real  and  apparent  motions  of  these  planets  have 
already  been  briefly  described  in  Part  I.,  Chapter  IY.  It 
will  be  remembered  that,  in  accordance  with  KEPLER'S 
third  law,  their  periods  of  revolution  around  the  sun  are 
less  than  that  of  the  earth.  Consequently  they  overtake 
the  latter  between  successive  inferior  conjunctions. 

The  interval  between  these  conjunctions  is  about  four 
months  in  the  case  of  Mercury,  and  between  nineteen  and 
twenty  months  in  that  of  Venus.  At  the  end  of  this 
period  each  repeats  the  same  series  of  motions  relative  to 
the  sun.  What  these  motions  are  can  be  readily  seen  by 
studying  Fig.  84.  In  the  first  place,  suppose  the  earth, 
at  any  point,  E,  of  its  orbit,  and  if  we  draw  a  line,  E  L 
or  EM,  from  E,  tangent  to  the  orbit  of  either  of  these 
planets,  it  is  evident  that  the  angle  which  this  line  makes 
with  that  drawn  to  the  sun  is  the  greatest  elongation  of 
the  planet  from  the  sun.  The  orbits  being  eccentric,  this 


ASPECTS  OF  MERCURY  AND   VENUS. 


311 


FIG.  84. 


elongation  varies  with  the  position  of  the  earth.  In  the 
case  of  Mercury  it  ranges  from  16°  to  29°,  while  in  the 
ease  of  Venus,  the  orbit  of  which  is  nearly  circular,  it 

varies  very  little  from 
45°.  These  planets, 
therefore,  seem  to  have 
an  oscillating  motion, 
first  swinging  toward  the 
east  of  the  sun,  and  then 
toward  the  west  of  it,  as 
already  explained  in  Part 
L,  Chapter  IY.  Since, 
owing  to  the  annual  revo- 
lution of  the  earth,  the 
sun  has  a  constant  east- 
ward motion  among  the 
stars,  these  planets  must 
have,  on  the  whole,  a  corresponding  though  intermittent 
motion  in  the  same  direction.  Therefore  the  ancient 
astronomers  supposed  their  period  of  revolution  to  be  one 
year,  the  same  as  that  of  the  sun. 

If,  again,  we  draw  a  line  ES  C  from  the  earth  through 
the  sun,  it  is  evident  that  the  first  point  /,  in  which  this 
line  cuts  the  orbit  of  the  planet,  or  the  point  of  inferior 
conjunction,  will  (leaving  eccentricity  out  of  the  question) 
be  the  least  distance  of  the  planet  from  the  earth,  while  the 
second  point  (7,  or  the  point  of 
superior  conjunction,  on  the  op- 
posite side  of  the  sun,  will  be 
the  greatest  distance.  Owing  to 
the  difference  of  these  distances, 
the  apparent  magnitude  of  these 

T  j.  .,  ,,          FIG.    85. — APPARENT    MAGNI- 

planets,  as  seen  irom  the  earth,  TUDES  OF  THE  DISK  OF 
is  subject  to  great  variations.  MERCURY. 

Fig.  85  shows  these  variations  in  the  case  of  Mercury, 
A  representing  its  apparent  magnitude  when  at  its  greatest 
distance,  B  when  at  its  mean  distance,  and  C  when  at  its 


312  ASTRONOMY. 

least  distance.  In  the  case  of  Venus  (Fig.  86)  the  varia- 
tions are  much  greater  than  in  that  of  Mercury,  the  great- 
est distance,  1-72,  being  more  than  six  times  the  least 
distance,  which  is  only  0  •  28.  The  variations  of  apparent 
magnitude  are  therefore  great  in  the  same  proportion. 

In  thus  representing  the  apparent  angular  magnitude 
of  these  planets,  we  suppose  their  whole  disks  to  be  visible, 
as  they  would  be  if  they  shone  by  their  own  light.  But 
since  they  can  be  seen  only  by  the  reflected  light  of  the 
sun,  only  those  portions  of  the  disk  can  be  seen  which  are 
at  the  same  time  visible  from  the  sun  and  from  the  earth. 
A  very  little  consideration  will  show  that  the  proportion 
of  the  disk  which  can  be  seen  constantly  diminishes  as  the 
planet  approaches  the  earth,  and  looks  larger. 


FlG.    86. — APPARENT  MAGNITUDES  OF  DISK  OP   VENUS. 

When  the  planet  is  at  its  greatest  distance,  or  in  superior 
conjunction  (C,  Fig.  84),  its  whole  illuminated  hemisphere 
can  be  seen  from  the  earth.  As  it  moves  around  and  ap- 
proaches the  earth,  the  illuminated  hemisphere  is  gradually 
turned  from  us.  At  the  point  of  greatest  elongation,  M 
or  Z,  one  half  the  hemisphere  is  visible,  and  the  planet 
has  the  form  of  the  moon  at  first  or  second  quarter.  As 
it  approaches  inferior  conjunction,  the  apparent  visible  disk 
assumes  the  form  of  a  crescent,  which  becomes  thinner 
and  thinner  as  the  planet  approaches  the  sun. 

Fig.  87  shows  the  apparent  disk  of  Mercury  at  various 
times  during  its  synodic  revolution.  The  planet  will  ap- 
pear brightest  when  this  disk  has  the  greatest  surface. 


ASPECTS  OF  MERCURY  AND    VENUS.  313 

This  occurs  about  half  way  between  greatest  elongation 
and  inferior  conjunction. 

In  consequence  of  the  changes  in  the  brilliancy  of  these 
planets  produced  by  the  variations  of  distance,  and  those 
produced  by  the  variations  in  the  proportion  of  illuminated 
disk  visible  from  the  earth,  partially  compensating  each 
other,  their  actual  brilliancy  is  not  subject  to  such  great 
variations  as  might  have  been  expected.  As  a  general  rule, 
Mercury  shines  with  a  light  exceeding  that  of  a  star  of 
the  first  magnitude.  But  owing  to  its  proximity  to  the 
sun,  it  can  never  be  seen  by  the  naked  eye  except  in  the 
west  a  short  time  after  sunset,  and  in  the  east  a  little  be- 
fore sunrise.  It  is  then  of  necessity  near  the  horizon,  and 


B  C 


cc 


FlG.  87. — APPEARANCE    OF  MERCURY  AT    DIFFERENT  POINTS    OF    ITS 

ORBIT. 

therefore  does  not  seem  so  bright  as  if  it  were  at  a  greater 
altitude.  In  our  latitudes  we  might  almost  say  that  it  is 
never  visible  except  in  the  morning  or  evening  twilight. 
In  higher  latitudes,  or  in  regions  where  the  air  is  less 
transparent,  it  is  scarcely  ever  visible  without  a  telescope. 
It  is  said  that  COPEKNICUS  died  without  ever  obtaining  a 
view  of  the  planet  Mercury. 

On  the  other  hand,  the  planet  Venus  is,  next  to  the  sun 
and  moon,  the  most  brilliant  object  in  the  heavens.  It  is 
so  much  brighter  than  any  fixed  star  that  there  can  seldom 
be  any  difficulty  in  identifying  it.  The  unpractised  ob- 
server might  under  some  circumstances  find  a  difficulty  in 


314  ASTRONOMY. 

distinguishing  between  Venus  and  Jupiter,  but  the  differ- 
ent motions  of  the  two  pfanets  will  enable  him  to  distin- 
guish them  if  they  are  watched  from  night  to  night  dur- 
ing several  weeks. 

§  2.    ASPECT  AND  ROTATION  OF  MERCURY. 

The  various  phases  of  Mercury,  as  dependent  upon  its 
various  positions  relative  to  the  sun,  have  already  been 
shown.  If  the  planet  were  an  opaque  sphere,  without  in- 
equalities and  without  an  atmosphere,  the  apparent  disk 
would  always  be  bounded  by  a  circle  on  one  side  and  an 
ellipse  on  the  other,  as  represented  in  the  figure. 
Whether  any  variation  from  this  simple  and  perfect  form 
has  ever  been  detected  is  an  open  question,  the  balance  of 
evidence  being  very  strongly  in  the  negative.  Since  no 
spots  are  visible  upon  it,  it  would  follow  that  unless  vari- 
ations of  form  due  to  inequalities  on  its  surface,  such  as 
mountains,  can  be  detected,  it  is  impossible  to  determine 
whether  the  planet  rotates  on  its  axis.  The  only  evidence 
in  favor  of  such  rotation  is  that  of  SCHROTER,  the  celebrated 
astronomer  of  Lilienthal,  who  made  the  telescopic  study 
of  the  moon  and  planets  his  principal  work.  About  the 
beginning  of  the  present  century  he  noticed  that  at  certain 
times  the  south  horn  of  the  crescent  of  Mercury  seemed 
to  be  blunted.  Attributing  this  appearance  to  the  shadow 
of  a  lofty  mountain,  he  concluded  that  the  planet  Mercury 
revolved  on  its  axis  in  a  little  more  than  24  hours.  But 
this  planet  has  since  been  studied  with  instruments  much 
more  powerful  than  those  of  SCHROTER,  and  no  confirma- 
tion of  his  results  has  been  obtained.  We  must  therefore 
conclude  that  the  period  of  rotation  of  Mercury  on  its 
axis  is  entirely  unknown. 

Respecting  an  atmosphere  of  Mercury,  the  evidence  is 
also  conflicting.  The  spectrum  of  this  planet  has  been 
studied  by  Dr.  YOGEL,  now  astronomer  at  the  Physical 
Observatory  of  Potsdam,  who  finds  that  its  principal  lines 


ASPECTS  OP  MERCURY  315 

coincide  with  those  of  the  sun.  Of  course  we  should 
expect  this  because  the  planet  shines  by  reflected  solar 
light.  But  he  also  finds  that  certain  lines  are  seen  in  the 
spectrum  of  Mercury  which  we  know  to  be  due  to  the  ab- 
sorption of  the  earth's  atmosphere,  and  which  appear 
more  dense  than  they  should  from  the  simple  passage 
through  our  atmosphere.  This  would  seem  to  show  that 
Mercury  has  an  envelope  of  gaseous  matter  somewhat  like 
our  own.  On  the  other  hand,  Dr.  ZOLLNER,  of  Leipsic, 
by  measuring  the  amount  of  light  reflected  by  the  planet 
at  various  times,  concludes  that  Mercury,  like  our  moon, 
is  devoid  of  any  atmosphere  sufficient  to  reflect  the  light 
of  the  sun.  We  may  therefore  regard  it  as  doubtful 
whether  any  evidence  of  an  atmosphere  of  Mercury  can 
be  obtained,  and  it  is  certain  that  we  know  nothing  defi- 
nite respecting  its  physical  constitution. 


§   3.    THE  ASPECT  AND   SUPPOSED  ROTATION    OF 

VENUS. 

As  Venus  sometimes  comes  nearer  the  earth  than  any 
other  primary  planet,  astronomers  have  examined  its  sur- 
face with  great  interest  ever  since  the  invention  of  the 
telescope.  But  no  conclusive  evidence  respecting  the  ro- 
tation of  the  planet  and  no  proof  of  any  changes  or  any 
inequalities  on  its  surface  have  ever  been  obtained.  The 
observations  are  either  very  discordant,  or  so  difficult 
and  unreliable  that  we  may  readily  suppose  the  ob-- 
servers  to  have  been  misled  as  to  what  they  saw.  In  1767 
CASSINI  thought  he  saw  a  bright  spot  on  Venus  during 
several  successive  evenings,  and  concluded,  from  his  sup- 
posed observation  that  the  planet  revolved  on  its  axis  in  a 
little  more  than  23  hours.  The  subject  was  next  taken  up 
by  BLANCHINI,  an  Italian  astronomer,  who  supposed  that 
he  saw  a  number  of  dark  regions  on  the  planet.  These  he 
considered  to  be  seas  or  oceans,  and  he  went  so  far  as  to 
give  them  names.  Watching  them  from  night  to  night, 


316  AST&ONOMY. 

he  concluded  that  the  time  of  rotation  of  Venus  was  more 
than  24  days.  Again,  SCOOTER  thought  that,  when  Ve- 
nus was  a  crescent,  one  of  its  sharp  points  was  blunted 
at  certain  intervals,  as  in  the  case  of  Mercury.  He  formed 
the  same  theory  of  the  cause  of  this  appearance —namely , 
that  it  was  due  to  the  shadow  of  a  high  mountain.  He  con- 
cluded that  the  time  of  rotation  found  by  CASSINI  was  near- 
ly correct.  Finally,  in  1842,  DE  Yico,  of  Rome,  thought 
he  could  see  the  same  dark  regions  or  oceans  on  the  planet 
which  had  been  seen  by  BLANCHINI.  He  concluded  that  the 
true  time  of  rotation  was  23h  21m  22s.  This  result  has  gone 
into  many  of  our  text-books  as  conclusive,  but  it  is  contra- 
dicted by  the  investigation  of  many  excellent  observers 
with  much  better  instruments.  HERSCHEL  was  never  able  to 
see  any  permanent  markings  on  Venus.  If  he  ever  caught 
a  glimpse  of  spots,  they  were  so  transient  that  he  could 
gather  no  evidence  respecting  the  rotation  of  the  planet. 
He  therefore  concluded  that  if  they  really  existed,  they 
were  due  entirely  to  clouds  floating  in  an  atmosphere,  and 
that  no  time  of  rotation  could  be  deduced  by  observing 
them.  This  view  of  HERSCHEL,  so  far  as  concerns  the 
aspect  of  the  planet,  is  confirmed  by  a  study  with  the  most 
powerful  telescopes  in  recent  times.  With  the  great 
Washington  telescope,  no  permanent  dark  spots  and  no 
regular  blunting  of  either  horn  has  ever  been  observed. 

It  may  seem  curious  that  skilled  observers  could  have 
been  deceived  as  to  what  they  saw  ;  but  we  must  remem- 
ber that  there  are  many  celestial  phenomena  which  are  ex- 
tremely difficult  to  make  out.  By  looking  at  a  drawing 
of  a  planet  or  nebula,  and  seeing  how  plain  every  thing 
seems  in  the  picture,  we  may  be  entirely  deceived  as  to  the 
actual  aspect  with  a  telescope.  Under  the  circumstances,  if 
the  observer  has  any  preconceived  theory,  it  is  very  easy 
for  him  to  think  he  sees  every  thing  in  accordance  with 
that  theory.  Now,  there  are  at  all  times  great  differences 
in  the  brilliancy  of  the  different  parts  of  the  disk  of  Venus. 
It  is  brightest  near  the  round  edge  which  is  turned 


ASPECTS  OF  VENUS.  317 

toward  the  sun.  Over  a  small  space  the  brightness  is  such 
that  some  recent  observers  have  formed  a  theory  that  the 
sun's  light  is  reflected  as  from  a  mirror.  On  the  other 
hand,  near  the  boundary  between  light  and  darkness,  the 
surface  is  much  darker.  Moreover,  owing  to  the  undu- 
lations of  our  atmosphere,  the  aspect  of  any  planet  so  small 
and  bright  as  Venus  is  constantly  changing.  The  only 
way  to  reach  any  certain  conclusion  respecting  its  ap- 
pearance is  to  take  an  average,  as  it  were,  of  the  appear- 
ances as  modified  by  the  undulations.  In  taking  this  aver- 
age, it  is  very  easy  to  imagine  variations  of  light  and  dark- 
ness which  have  no  real  existence  ;  it  is  not,  therefore,  sur- 
prising that  one  astronomer  should  follow  in  the  footsteps 
of  another  in  seeing  imaginary  markings. 

Atmosphere  of  Venus. — The  evidence  of  an  atmosphere 
of  Venus  is  perhaps  more  conclusive  than  in  the  case  of 
any  other  planet.  When  Venus  is  observed  very  near 
its  inferior  conjunction,  and  when  it  therefore  presents  the 
view  of  a  very  thin  crescent,  it  is  found  that  this  crescent 
extends  over  more  than  180°.  This  would  be  evidently 
impossible  unless  the  sun  illuminated  more  than  one  half 
the  planet  One  of  the  most  fortunate  observers  of  this 
phenomenon  was  Prof essor  C.  S.  LYMAN,  of  Yale  College, 
who  observed  Venus  in  December,  1866.  The  inferior 
conjunction  of  the  planet  occurred  near  the  ascending 
node,  so  that  its  angular  distance  from  the  sun  was  less 
than  it  had  been  at  any  former  time  during  the  present  cen- 
tury. Professor  LYMAN  saw  the  disk,  not  as  a  thin  cres- 
cent, but  as  an  entire  and  extremely  fine  circle  of  light. 
We  therefore  conclude  that  Venus  has  an  atmosphere 
which  exercises  so  powerful  a  refraction  upon  the  light  of 
the  sun  that  the  latter  illuminates  several  degrees  more 
than  one  half  the  globe.  A  phenomenon  which  must  be 
attributed  to  the  same  cause  has  several  times  been  ob- 
served during  transits  of  Venus.  During  the  transit  of 
December  8th,  1874,  most  of  the  observers  who  enjoyed 
a  fine  steady  atmosphere  saw  that  when  Venus  was  par- 


318  ASTRONOMY. 

tially  projected  on  the  sun,  ^the  outline  of  that  part  of  its 
disk  outside  the  sun  could  be  distinguished  by  a  delicate 
line  of  light.  A  similar  appearance  was  noticed  by  DAVID 
RITTENHOUSE,  of  Philadelphia,  on  June  3d,  1769.  From 
these  several  observations,  it  would  seem  that  the  refractive 
power  of  the  atmosphere  of  Venus  is  greater  than  that  of 
the  earth.  Attempts  have  been  made  to  determine  its  ex- 
act amountj  but  they  are  too  uncertain  to  be  worthy  of 
quotation. 

§  4.  TRANSITS  OP  MERCURY  AND  VENUS. 

When  Mercury  or  Venus  passes  between  the  earth  and 
sun,  so  as  to  appear  projected  on  the  sun's  disk,  the  phe- 
nomenon is  called  a  transit.  If  these  planets  moved  around 
the  sun  in  the  plane  of  the  ecliptic,  it  is  evident  that 
there  would  be  a  transit  at  every  inferior  conjunction.  But 
since  their  orbits  are  in  reality  inclined  to  the  ecliptic, 
transits  can  occur  only  when  the  inferior  conjunction  takes 
place  near  the  node.  In  order  that  there  may  be  a  transit, 
the  latitude  of  the  planet,  as  seen  from  the  earth,  must 
be  less  than  the  angular  semi- diameter  of  the  sun — that  is, 
less  than  16'.* 

The  longitude  of  the  descending  node  of  Mercury  at  the 
present  time  is  227°,  and  therefore  that  of  the  ascending 
node  47°.  The  earth  has  these  longitudes  on  May  7th  and 
November  9th.  Since  a  transit  can  occur  only  within  a 
few  degrees  of  a  node,  Mercury  can  transit  only  within  a 
few  days  of  these  epochs. 

The  longitude  of  the  descending  node  of  Venus  is  now 

*  The  mathematical  student,  know  ing  that  the  inclination  of  the  orbit 
of  Mercury  is  7°  0'  and  that  of  Venus  3°  24',  will  find  it  an  interesting 
problem  to  calculate  the  limits  of  distance  from  the  node  within  which  in- 
ferior conjunction  must  take  place  in  order  that  a  transit  may  occur. 
From  the  geocentric  latitude  16'  the  heliocentric  latitude  may  be  found 
by  multiplying  by  the  distance  from  the  earth  and  dividing  by  that  from 
the  sun.  He  will  find  these  limits  to  be  a  little  greater  for  Mercury 
than  for  Venus,  notwithstanding  its  greater  inclination,  and  to  be  only 
a  few  degrees  in  either  case, 


TRANSITS  OF  MERCURY.  319 

about  256°,  and  therefore  that  of  the  ascending  node  is 
76°.  The  earth  has  these  longitudes  on  June  6th  and  De- 
cember 7th  of  each  year.  Transits  of  Venus  can  there- 
fore occur  only  within  two  or  three  days  of  these  times. 

Recurrence  of  Transits  of  Mercury.— The  transits  of  Mer- 
cury and  Venus  recur  in  cycles  which  resemble  the  eighteen- 
year  cycle  of  eclipses,  but  in  which  the  precision  of  the  recurrence 
is  less  striking.  From  the  mean  motions  of  Mercury  and  the  earth 
already  given,  we  find  that  the  mean  synodic  period  of  Mercury  is, 
in  decimals  of  a  Julian  year,  0^-317256.  Three  synodic  periods  are 
therefore  some  eighteen  days  less  than  a  year.  If,  then,  we  suppose 
an  inferior  conjunction  of  Mercury  to  occur  exactly  at  a  node,  the 
third  conjunction  following  will  take  place  about  eighteen  days 
before  the  earth  again  reaches  the  node,  and  therefore  about  18° 
from  the  node,  since  the  earth  moves  nearly  1°  in  a  day.  This  is 
far  outside  the  limit  of  a  transit  ;  we  must,  therefore,  wait  until 
another  conjunction  occurs  near  the  same  place.  To  find  when 
this  will  be,  the  successive  vulgar  fractions  which  converge  toward 
the  value  of  the  above  period  may  be  found  by  the  method  of  con- 
tinued fractions.  The  first  five  of  these  fractions  are  : 

*      T"T      A      H      A% 

Here  the  denominators  are  numbers  of  synodic  periods,  while  the 
numerators  are  the  approximate  corresponding  number  of  years. 
By  actual  multiplication  we  find  : 

3  Periods  =  0^-951768  =  U- -048232.  Error  =-  17° 

19       "        =  6-027864=  6+ -027864.         "      =  +  10° 

22      "        =  6-979632=  7- -020368.         "             -    7° 

.41      "        =  13-007496=  13+  -007496.         "      =  +    2°.  7 

145      "        =  46-002120=  46+ -002120.        "      =  +    0°-76 

In  this  table  the  errors  show  the  number  of  degrees  from  the 
node  at  which  the  inferior  conjunction  will  occur  at  the  end  of  one 
year,  six  years,  seven  years,  etc.  They  are  found  by  multiplying 
the  fraction  by  which  the  intervals  exceed  or  fall  short  of  an  entire 
number  of  years  by  360°.  It  will  be  seen  that  the  19th,  22d,  41st, 
and  145th  conjunctions  occur  nearer  and  nearer  the  node,  or,  sup- 
posing that  we  do  not  start  from  a  node,  nearer  and  nearer  the  point 
of  the  orbits  from  which  we  do  start.  It  follows  that  the  recur- 
rence of  a  transit  of  Mercury  at  the  same  node  is  possible  at  the 
end  of  7  years,  probable  at  the  end  of  13  years,  and  almost  certain 
at  the  end  of  46  years.  The  latter  is  the  cycle  which  it  would  be 
most  convenient  to  take  as  that  in  which  all  the  transits  would 
recur,  but  it  would  still  not  be  so  exact  as  the  eclipse  cycle  of  18 
years  11  days. 


320 


ASTRONOMY. 


The  following  table  shows  the  dates  of  occurrence  of  transits  of 
Mercury  during  the  present  century.  They  are  separated  into  May 
transits,  which  occur  near  the  'descending  node,  and  November 
ones,  which  occur  near  the  ascending  node.  November  transits  are 
the  most  numerous,  because  Mercury  is  then  nearer  the  sun,  and 
the  transit  limits  are  wider. 


1799,  May  6. 
1832,  May  5. 
1845,  May  8. 
1878,  May  6. 
1891,  May  9. 


1802, 
1815, 
1822, 
1835, 

1848, 
1861, 
1868, 
1881, 
1894, 


Nov.  9. 
Nov.  11. 
Nov.  5. 
Nov.  7. 
Nov.  10. 
Nov.  12. 
Nov.  5. 
Nov.  7. 
Nov.  10. 


It  will  be  seen  that  in  a  cycle  of  46  years  there  are  two  May  tran- 
sits and  four  November  ones,  so  that  the  latter  are  twice  as  nu- 
merous as  the  former.  These  numbers  may,  however,  change  slightly 
at  some  future  time  through  the  failure  of  a  recurrence,  or  the  en- 
trance of  a  new  transit  into  the  series.  Thus,  in  the  May  series,  it 
is  doubtful  whether  there  will  be  an  actual  transit  46  years  after 
1891 — that  is,  in  1937 — or  whether  Mercury  will  only  pass  very  near 
the  limb  of  the  sun.  On  the  other  hand,  Mercury  passed  within  a  few 
minutes  of  the  sun's  limb  on  May  3d,  1865,  and  it  will  probably 
graze  the  limb  46  years  later — that  is,  on  May  4th  or  5th,  1911. 

Recurrence  of  Transits  of  Venus.— For  many  centuries 
past  and  to  come,  transits  of  Venus  occur  in  a  cycle  more  exact  than 
those  of  Mercury.  It  happens  that  eight  times  the  mean  motion  of 
Venus  is  very  nearly  the  same  as  thirteen  times  the  mean  motion 

of  the  earth  ;  in  other  words,  Venus 
makes  13  revolutions  around  the 
sun  in  nearly  the  same  time  that 
the  earth  makes  8  revolutions — 
that  is,  in  eight  years.  During 
this  period  there  will  be  5  inferior 
conjunctions  of  Venus,  because  the 
latter  has  made  5  revolutions  more 
than  the  earth.  Consequently,  if 
we  wait  eight  years  from  an  inferior 
conjunction  of  Venus,  we  shall,  at 
the  end  of  that  time,  have  another 
inferior  conjunction,  the  fifth  in 
regular  order,  at  nearly  the  same 
point  of  the  two  orbits.  It  will, 
therefore,  occur  at  the  same  time 
of  the  year,  and  in  nearly  the  same 
position  relative  to  the  node  of  Venus.  In  Fig.  83  let  8  represent 
the  sun,  and  the  circle  drawn  around  it  the  orbit  of  the  earth. 


FIG.  88. — CONJUNCTIONS  OF 

VENUS. 


TRANSITS  OF  VENUS.  321 

Suppose  also  that  at  the  moment  of  the  inferior  conjunction  of 
Venus,  we  draw  a  straight  line  S 1  through  Venus  to  the  earth  at  1. 
We  shall  then  have  to  wait  about  If  years  for  another  inferior  con- 
junction, during  which  time  the  earth  will  have  made  one  revolu- 
tion and  f  of  another,  and  Venus  2f  revolutions.  The  straight  line 
drawn  through  the  point  of  inferior  conjunction  will  then  be  S  2. 
The  third  conjunction  will  in  the  same  way  take  place  in  the  posi- 
tion 8  3,  which  is  If  revolutions  further  advanced  ;  the  fourth  in 
the  position  8  4,  and  the  fifth  in  the  position  8  5.  If  the  corre- 
spondence of  the  motions  were  exact,  the  sixth  conjunction,  at  the 
end  of  8  years  (5  x  If  =  8),  would  again  take  place  in  the  original 
position  8  1,  and  all  subsequent  ones  would  follow  in  the  same 
order.  All  inferior  conjunctions  would  then  take  place  at  one  of 
these  five  points,  and  no  transit  would  ever  be  possible  unless  one 
of  these  points  should  chance  to  be  very  near  the  line  of  nodes. 

In  fact,  however,  the  correspondence  is  not  perfectly  exact,  but, 
at  the  end  of  8  years,  the  sixth  conjunction  will  take  place  not 
exactly  along  the  line  $1,  but  a  little  before  the  two  bodies  reach 
this  line.  The  actual  angle  between  the  line  8 1  and  that  of  the 
sixth  conjunction  will  be  about  2°  22',  the  point  shifting  back  to- 
ward the  direction  $4.  Of  course,  each  following  conjunction  will 
take  place  at  the  same  distance  back  from  that  of  eight  years  before, 
leaving  out  small  changes  due  to  the  eccentricities  of  the  orbits  and 
the  variations  of  their  elements.  It  follows  then  that  if  we  suppose 
the  five  lines  of  conjunction  to  have  a  retrograde  motion  in  a 
direction  the  opposite  of  that  of  the  arrow,  amounting  to  2°  22'  in 
eight  years,  all  the  inferior  conjunctions  will  take  place  along  these 
five  lines.  The  distance  apart  of  the  lines  being  72°  and  the 
motion  about  18'  per  year,  the  intervals  between  the  passages  of 
the  several  conjunction  lines  over  the  line  of  nodes  will  be  about 
240  years.  Really,  the  exact  time  is  243  years. 

Suppose,  now,  that  a  conjunction  should  take  place  exactly  at  a 
node,  then  the  fifth  following  conjunction  would  take  place 
2°  22'  before  reaching  the  node.  The  limits  within  which  a  transit 
can  occur  are,  however,  only  1°  46'  on  each  side  of  the  node  ;  con- 
sequently, there  would  be  no  further  transit  at  that  node  until  the 
next  following  conjunction  point  reached  it,  which  would  happen  at 
the  end  of  243  years.  If,  however,  the  conjunction  should  take  place 
between  0°  36'  and  1°  46'  after  reaching  the  node,  there  would  be  a 
transit,  and  the  fifth  following  conjunction  would  also  occur  within 
the  limit  on  the  other  side  of  the  node,  so  that  we  should  have  two 
transits  eight  years  apart.  We  may,  therefore,  have  either  one 
transit  or  two  according  to  the  distance  from  the  node  at  which  the 
first  transit  occurs.  We  thus  have  at  any  one  node  either  a  single 
transit,  or  a  pair  of  transits  eight  years  apart,  in  a  cycle  of  243  years. 
At  the  middle  of  this  cycle  the  node  will  be  half  way  between  two 
of  the  conjunction  points — the  points  1  and  3,  for  instance  ;  but  it  is 
evident  that  in  this  case  the  opposite  node  will  coincide  with  the 
conjunction  point  2,  since  there  is  an  odd  number  of  such  points. 
It  follows,  therefore,  that  about  the  middle  of  the  interval  between 
two  consecutive  sets  of  transits  at  one  node  we  shall  have  a  transit 
or  a  pair  of  transits  at  the  opposite  node. 


322  ASTRONOMY. 

The  earth  passes  through  the  line  of  the  descending  node  of  the 
orbit  of  Venus  early  in  June  of  each  year,  and  through  the  ascending 
node  early  in  December.  It  follows,  therefore,  that  the  series  will 
be  a  transit  or  a  pair  of  transits  in  June  ;  then  an  interval  of  about  120 
years,  to  be  followed  by  a  transit  or  a  pair  of  transits  in  December, 
and  so  on.  Owing  to  the  eccentricity  of  the  orbits,  the  intervals 
will  not  be  exactly  equal,  the  motions  of  the  several  conjunction 
points  not  being  uniform,  nor  their  distance  exactly  72°.  The 
dates  and  intervals  of  the  transits  for  three  cycles  nearest  to  the 
present  time  are  as  follows  : 

Intervals. 
1518,  June  2.  1761,  June  5.  2004,  June  8.  8   years. 

1526,  June  1.  1769,  June  3.  2012,  June  6.  105*     " 

1631,  Dec.  7.  1874,  Dec.  9.  2117,  Dec.  11.  8       " 

1639,  Dec.  4.  1882,  Dec.  6.  2125,  Dec.  8.  12H     " 

The  243-year  cycle  is  so  exact  that  the  actual  deviations  from  it 
are  due  almost  entirely  to  the  secular  variation  of  the  orbits  of 
Vetius  and  the  Earth.  Moreover,  the  conjunction  of  December  8th, 
1874,  took  place  1°  25'  past  the  ascending  node,  so  that  the  con- 
junction of  1882  takes  place  about  1°  4'  before  reaching  the  node. 
Owing  to  the  near  approach  of  the  period  to  exactness,  several  pairs 
of  transits  near  this  node  have  taken  place  in  the  past,  at  equal  in- 
tervals of  243  years,  and  will  be  repeated  for  three  or  four  cycles  in 
the  future. 

Nearly  the  same  remark  applies  to  those  which  take  place  at  the 
descending  node,  where  pairs  of  transits  eight  years  apart  will 
occur  for  about  three  cycles  in  the  future.  Owing,  however,  to 
secular  variations  of  the  orbit,  the  conjunction  point  for  the  second 
June  transit  of  each  pair  and  the  first  December  transit  will,  after 
perhaps  a  thousand  years,  take  place  so  far  from  the  node  that  the 
planet  will  not  quite  touch  the  sun,  and  then  during  a  period  of 
many  centuries  there  will  only  be  one  transit  at  each  node  in 
every  243  years,  instead  of  two,  as  at  present. 


§  5.     SUPPOSED  INTRAMERCURIAL  PLANETS. 

Some  astronomers  are  of  opinion  that  there  is  a  small 
planet  or  a  group  of  planets  revolving  around  the  sun 
inside  the  orbit  of  Mercury.  To  this  supposed  planet  the 
name  Vulcan  has  been  given  ;  but  astronomers  generally 
discredit  the  existence  of  such  a  planet  of  considerable 
size,  because  the  evidence  in  its  favor  is  not  regarded  as 
conclusive. 


THE  SUPPOSED   VULCAN.  323 

The  evidence  in  favor  of  the  existence  of  such  planets  may  be 
divided  into  three  classes,  as  follows,  which  will  be  considered  in 
their  order  : 

(1)  A  motion  of  the  perihelion  of  the  orbit  of  Mercury,  supposed 
to  be  due  to  the  attraction  of  such  a  planet  or  group  of  planets. 

(2)  Transits  of  dark  bodies  across  the  disk  of  the  sun  which  have 
been  supposed  to  be  seen  by  various  observers  during  the  past  cen- 
tury. 

(3)  The  observation  of  certain  unidentified  objects  by  Professor 
WATSON  and  Mr.  LEWIS  SWIFT  during  the  total  eclipse  of  the  sun, 
July  29th,  1878. 

(1)  In  1858,  LE  VERRIER  made  a  careful  collection  of  all  the  obser- 
vations on  the  transits  of  Mercury  which  had  been  recorded  since  the 
invention  of  the  telescope.     The  result  of  that  investigation  was 
that  the  observed  times  of  transit  could  not  be  reconciled  with  the 
calculated  motion  of  the  planet,  as  due  to  the  gravitation  of  the 
other  bodies  of  the  solar  system.     He  found,  however,  that  if,  in 
addition  to  the  changes  of  the  orbit  due  to  the  attraction  of  the 
other  planets,  he  supposed  a  motion  of  the  perihelion  amounting  to 
36"  in  a  century,  the  observations  could  all  be    satisfied.     Such 
a  motion  might  be  produced  by  the  attraction  of    an  unknown 
planet  inside   the   orbit   of  Mercury.      Since,    however,    a  single 
planet,  in  order  to  produce  this  effect,  would  have  to  be  of  consid- 
erable size,  and  since  no  such  object  had  ever  been  observed  during 
a  total  eclipse  of  the  sun,  he  concluded  that  there  was  probably  a 
group  of  planets  much  too  small  to  be  separately  distinguished. 
So  far  as  the  discrepancy  between  theory  and  observation  is  con- 
cerned, these  results  of  LE  VERRIER' s  have  been  completely  con- 
firmed by  the  mathematical  researches  of  Mr.  G.  W.  HILL,  and  by 
observations  of  transits  since  LE  VERRIER'S  calculations  were  com- 
pleted.    Indeed,  the  result  of  these  researches  and  observations  is 
that  the  motion  of  the  perihelion  is  even  greater  than  that  found 
by  LE  VERRIER,  the  surplus  motion  being  more  than  40"  in  a  cen- 
tury.    There  is  no  known  way  of  accounting  for  this  motion  in 
accordance  with  well-established  laws,  except  by  supposing  matter 
of  some  sort  to  be  revolving  around  the  sun  in  the  supposed  posi- 
tion.    At  the  same  time  it  is  always  possible  that  the  effect  may 
be  produced  by  some  unknown  cause.* 

(2)  Astronomical  records  contain    upward  of  twenty  instances 
in  which  dark  bodies  have  been  supposed  to  be  seen  in  transit 
across  the  disk  of  the  sun.     If  we  suppose  these  observations  to  be 
all  perfectly  correct,  the  existence  of  a  great  number  of  considerable 
planets  within  the  orbit  of  Mercury  would  be  placed  beyond  doubt. 
But  a  critical  analysis  shows  that  these  observations,  considered  as  a 
class,  are  not  entitled  to  the  slightest  credence.     In  the  first  place, 

*  An  electro-dynamic  theory  of  attraction  has  been  within  the  past 
twenty  years  suggested  by  several  German  physicists,  which  involves 
a  small  variation  from  the  ordinary  theory  of  gravitation.  It  has  been 
shown  that,  by  supposing  this  theory  true,  the  motion  of  the  perihelion 
of  Mercury  could  be  accounted  for  by  the  attraction  of  the  sun. 


324  ASTRONOMY. 

scarcely  any  of  them  were  made  by  experienced  observers  with 
powerful  instruments.  It  is  very  easy  for  an  unpractised  observer 
to  mistake  a  round  solar  spot  for  a  planet  in  transit.  It  may  there- 
fore be  supposed  that  in  many  cases  the  observer  saw  nothing  but 
a  spot  on  the  sun.  In  fact,  the  very  last  instance  of  the  kind  on 
record  was  an  observation  by  WEBER  at  Peckeloh,  on  April  4th, 
1876.  He  published  an  account  of  his  observation,  which  he  sup- 
posed was  that  of  a  planet,  but  when  the  publication  reached  other 
observers,  who  had  been  examining  the  sun  at  the  same  time,  it 
was  shown  conclusively  that  what  he  saw  was  nothing  more  than 
an  unusually  round  solar  spot.  Again,  in  most  of  the  cases  referred 
to,  the  object  seen  was  described  as  of  such  magnitude  that  it 
could  not  fail  to  have  been  noticed  during  total  eclipses  if  it  had 
any  real  existence.  It  is  also  to  be  noted  that  if  such  planets  ex- 
isted they  would  frequently  pass  over  the  disk  of  the  sun.  Dur- 
ing the  past  fifty  years  the  sun  has  been  observed  almost  every 
day  with  the  greatest  assiduity  by  eminent  observers,  armed  with 
powerful  instruments,  who  have  made  the  study  of  the  sun's  sur- 
face and  spots  the  principal  work  of  their  lives.  None  of  these 
observers  has  ever  recorded  the  transit  of  an  unknown  planet.  This 
evidence,  though  negative  in  form,  is,  under  the  circumstances,  con- 
clusive against  the  existence  of  such  a  planet  of  such  magnitude 
as  to  be  visible  in  transit  with  ordinary  instruments, 

(3)  The  observations  of  Professor  WATSON  during  the  total 
eclipse  above  mentioned  seem  to  afford  the  strongest  evidence  yet 
obtained  in  favor  of  the  real  existence  of  the  planet.  His  mode  of 
proceeding  was  briefly  this  :  Sweeping  to  the  west  of  the  sun 
during  the  eclipse,  he  saw  two  objects  in  positions  where,  suppos- 
ing the  pointing  of  his  telescope  accurately  known,  no  fixed  star 
existed.  There  is,  however,  a  pair  of  known  stars,  one  of  which  is 
about  a  degree  distant  from  one  of  the  unknown  objects,  and  the 
other  about  the  same  distance  and  direction  from  the  second.  It 
is  considered  by  some  that  Professor  WATSON'S  supposed  planets 
may  have  been  this  pair  of  stars.  Still,  if  Professor  WATSON'S 
planets  were  capable  of  producing  the  motion  of  the  perihelion  of 
Mercury  alread}r  referred  to,  we  should  regard  their  existence  as 
placed  beyond  reasonable  doubt.  But  his  observations  and  th». 
theoretical  results  of  LE  VERRIER  do  not  in  any  manner  strengthen 
each  other,  because,  if  we  suppose  the  observed  perturbations  in 
the  orbit  of  Mercury  to  be  due  to  planets  so  small  as  those  seen  by 
WATSON,  the  number  of  these  planets  must  be  many  thousands. 
Now,  it  is  very  certain  that  there  are  not  thousands  of  planets 
there  brighter  than  the  sixth  magnitude,  because  they  would  have 
been  seen  by  other  telescopes  engaged  in  the  same  search.  The 
smaller  we  suppose  the  individual  planets,  the  more  numerous  they 
must  be,  and,  finally,  if  we  consider  them  as  individually  invisible, 
they  will  probably  be  numbered  by  tens  of  thousands.  The  smaller 
and  more  numerous  they  are,  supposing  their  combined  mass  the 
same,  the  greater  the  sum  total  of  light  they  would  reflect.  At  a 
certain  point  the  amount  of  light  would  become  so  considerable 
that  the  group  would  appear  as  a  cloud-like  mass.  Now,  there  is 


THE  SUPPOSED   VULCAN.  325 

a  phenomenon  known  as  the  zodiacal  light,  which  is  probably  caused 
by  matter  either  in  a  gaseous  state  or  composed  of  small  particles  re- 
volving around  the  sun  at  various  distances  from  it.  This  light 
can  be  seen  rising  like  a  pillar  from  the  western  horizon  on  any 
very  clear  night  in  the  winter  or  spring.  Of  its  nature  scarcely 
any  thing  is  yet  known.  The  spectroscopic  observations  of  Pro- 
fessor WKIGHT,  of  Yale  College,  seem  to  indicate  that  it  is  seen  by 
reflected  sunlight.  Very  different  views,  however,  have  obtained 
respecting  its  constitution,  and  even  its  position,  some  having  held 
that  it  is  a  ring  surrounding  the  earth.  We  can  therefore  merely 
suggest  the  possibility  that  the  observed  motion  of  the  perihelion 
of  Mercury  is  produced  by  the  attraction  of  this  mass. 


CHAPTER  IV. 

THE    MOON. 

IN  Chapter  VII.  of  the  preceding  part  we  have  de- 
scribed the  motions  of  the  moon  and  its  relation  to  the 
earth.  We  shall  now  explain  its  physical  constitution  as 
revealed  by  the  telescope. 

When  it  became  clearly  understood  that  the  earth  and 
moon  were  to  be  regarded  as  bodies  of  one  class,  and  that 
the  old  notion  of  an  impassable  gulf  between  the  character 
of  bodies  celestial  and  bodies  terrestrial  was  unfounded, 
the  question  whether  the  moon  was  like  the  earth  in  all  its 
details  became  one  of  great  interest.  The  point  of  most 
especial  interest  was  whether  the  moon  could,  like  the 
earth,  be  peopled  by  intelligent  inhabitants.  Accordingly, 
when  the  telescope  was  invented  by  GALILEO,  one  of  the 
first  objects  examined  was  the  moon.  With  every  im- 
provement of  the  instrument,  the  examination  became 
more  thorough,  so  that  the  moon  has  been  an  object  of 
careful  study  by  the  physical  astronomer. 

The  immediate  successors  of  GALILEO  thought  that  they 
perceived  the  surface  of  the  moon,  like  that  of  our  globe, 
to  be  diversified  with  land  and  water.  Certain  regions  ap- 
peared dark  and,  for  the  most  part,  smooth,  while  others 
were  bright  and  evidently  broken  up  into  hills  and  valleys. 
The  former  regions  were  supposed  to  be  oceans,  and  re- 
ceived names  to  correspond  with  this  idea.  These  names 
continue  to  the  present  day,  although  we  now  know  that 
there  are  no  oceans  there. 

With  every  improvement  in  the  means  of  research,  it 


THE  MOON.  327 

has  become  more  and  more  evident  that  the  surface  of  the 
moon  is  totally  unlike  that  of  our  earth.  There  are  no 
oceans,  seas,  rivers,  air,  clouds,  or  vapor.  We  can  hardly 
suppose  that  animal  or  vegetable  life  exists  under  such 
circumstances,  the  fundamental  conditions  of  such  ex- 
istence on  our  earth  being  entirely  wanting.  We  might 
almost  as  well  suppose  a  piece  of  granite  or  lava  to  be  the 
abode  of  life  as  the  surface  of  the  moon  to  be  such. 

Before  proceeding  with  a  description  of  the  lunar  sur- 
face, as  made  known  to  us  by  the  telescopes  of  the  present 
time,  it  will  be  well  to  give  some  estimates  of  the  visi- 
bility of  objects  on  the  moon  by  means  of  our  instruments. 
Speaking  in  a  rough  way,  we  may  say  that  the  length  of 
one  mile  on  the  moon  would,  as  seen  from  the  earth,  sub- 
tend an  angle  of  1"  of  arc.  More  exactly,  the  angle  sub- 
tended would  range  between  0"-8  and  0"-9,  according  to 
the  varying  distance  of  the  moon.  In  order  that  an  ob- 
ject may  be  plainly  visible  to  the  naked  eye,  it  must  sub- 
tend an  angle  of  nearly  V.  Consequently,  a  magnifying 
power  of  60  is  required  to  render  a  round  object  one  mile 
in  diameter  on  the  surface  of  the  moon  plainly  visible. 
Starting  from  this  fact,  we  may  readily  form  the  follow- 
ing table,  showing  the  diameters  of  the  smallest  objects 
that  can  be  seen  with  different  magnifying  powers,  always 
assuming  that  vision  with  these  powers  is  perfect : 

Power      60  ;  diameter  of  object  1  mile. 

Power    150  ;  diameter  2000  feet. 

Power    500  ;  diameter  600  feet. 

Power  1000  ;  diameter  300  feet. 

Power  2000  ;  diameter  150  feet. 

If  telescopic  power  could  be  increased  indefinitely,  there 
would  of  course  be  no  limit  to  the  minuteness  of  an  ob- 
ject visible  on  the  moon's  surface.  But  the  necessary 
imperfections  of  all  telescopes  are  such  that  only  in  extra- 
ordinary cases  can  any  thing  be  gained  by  increasing  the 


328  ASTRONOMY. 

magnifying  power  beyond  1000.  The  influence  of  warm 
and  cold  currents  in  our  atmosphere  is  such  as  will  for- 
ever prevent  the  advantageous  use  of  high  magnifying 
powers.  After  a  certain  limit  we  see  nothing  more  by 
increasing  the  power,  vision  becoming  indistinct  in  pro- 
portion as  the  power  is  increased.  It  may  be  doubted 
whether  the  moon  was  ever  seen  through  a  telescope  to  so 
good  advantage  as  she  would  be  seen  with  a  magnifying 
power  of  500,  unaccompanied  by  any  drawback  from  at- 
mospheric vibrations  or  imperfection  of  the  telescope. 
In  other  words,  it  is  hardly  likely  that  an  object  less  than 
600  feet  in  extent  could  ever  be  seen  on  the  moon  by  any 
telescope  whatever,  unless  it  were  possible  to  mount  the 
instrument  above  the  atmosphere  of  the  earth.  It  is  there- 
fore only  the  great  features  on  the  surface  of  the  moon, 
and  not  the  minute  ones,  which  can  be  made  out  with  the 
telescope. 

Character  of  the  Moon's  Surface.  — The  most  striking 
point  of  difference  between  the  earth  and  moon  is  seen  in 
the  total  absence  from  the  latter  of  any  tiling  that  looks 
like  an  undulating  surface.  No  formations  similar  to  our 
valleys  and  mountain-chains  have  been  detected.  The 
lowest  surface  of  the  moon  which  can  be  seen  with  the 
telescope  appears  to  be  nearly  smooth  and  flat,  or,  to 
speak  more  exactly,  spherical  (because  the  moon  is  a 
sphere).  This  surface  has  different  shades  of  color  in 
different  regions.  Some  portions  are  of  a  bright,  silvery 
tint,  while  others  have  a  dark  gray  appearance.  These  dif- 
ferences of  tint  seem  to  arise  from  differences  of  material. 

Upon  this  surface  as  a  foundation  are  built  numerous 
formations  of  various  sizes,  but  all  of  a  very  simple  char- 
acter. Their  general  form  can  be  made  out  by  the  aid  of 
Fig.  89,  and  their  dimensions  by  the  scale  of  miles  at 
the  bottom  of  it.  The  largest  and  most  prominent 
features  are  known  as  craters.  They  have  a  typical  form 
consisting  of  a  round  or  oval  rugged  wall  rising  from  the 
plane  in  the  mariner  of  a  circular  fortification.  These 


THE  MOON'S  SURFACE. 


walls  are  frequently  from  three  to  six  thousand  metres  in 
height,  very  rough  and  broken.     In  their  interior  we  see 


FlG.    89. — ASPECT   OF   THE   MOON'S   SURFACE. 

the  plane  surface  of  the  moon  already  described.     It  is, 
however,  generally  covered  with  fragments  or  broken  up 


330  ASTRONOMY. 

by  small  inequalities  so  as  not  to  be  easily  made  out.  In 
the  centre  of  the  craters  we  .frequently  find  a  conical  for- 
mation rising  up  to  a  considerable  height,  and  much  larger 
than  the  inequalities  just  described.  In  the  craters  we 
have  a  vague  resemblance  to  volcanic  formations  upon  the 
earth,  the  principal  difference  being  that  their  magnitude 
is  very  much  greater  than  any  thing  known  here.  The 
diameter  of  the  larger  ones  ranges  from  50  to  200  kilo- 
metres, while  the  smallest  are  so  minute  as  to  be  hardly 
visible  with  the  telescope. 

When  the  moon  is  only  a  few  days  old,  the  sun's  rays 
strike  very  obliquely  upon  the  lunar  mountains,  and  they 
cast  long  shadows.  From  the  known  position  of  the  sun, 
moon,  and  earth,  and  from  the  measured  length  of  these 
shadows,  the  heights  of  the  mountains  can  be  calculated. 
It  is  thus  found  that  some  of  the  mountains  near  the  south 
pole  rise  to  a  height  of  8000  or  9000  metres  (from  25,000 
to  30,000  feet)  above  the  general  surface  of  the  moon. 
Heights  of  from  3000  to  YOOO  metres  are  very  common 
over  almost  the  whole  lunar  surface. 

Next  to  the  so-called  craters  visible  on  the  lunar  disk, 
the  most  curious  features  are  certain  long  bright  streaks, 
which  the  Germans  call  rills  or  furrows.  These  extend 
in  long  radiations  over  certain  of  the  craters,  and  have  the 
appearance  of  cracks  in  the  lunar  surface  which  have  been 
subsequently  filled  by  a  brilliant  white  material.  NA- 
SMYTH  and  CARPENTER  have  described  some  experiments 
designed  to  produce  this  appearance  artificially.  They 
took  hollow  glass  globes,  filled  them  with  water,  and  heat- 
ed them  until  the  surface  was  cracked.  The  cracks  gen- 
erated at  the  weakest  point  of  the  surface  radiate  from  the 
point  in  a  manner  strikingly  similar  in  appearance  to  the 
rills  on  the  moon.  It  would,  however,  be  premature  to 
conclude  that  the  latter  were  actually  produced  in  this 
way. 

The  question  of  the  origin  of  the  lunar  features  has  a 
bearing  on  theories  of  terrestrial  geology  as  well  as  upon 


LIGHT  AND  HEAT  OF  THE  MOON.  331 

various  questions  respecting  the  past  history  of  the  moon 
itself.  It  has  been  considered  in  this  aspect  by  various 
geologists. 

Lunar  Atmosphere. — The  question  whether  the  moon 
has  an  atmosphere  has  been  much  discussed.  The  only 
conclusion  which  has  yet  been  reached  is  that  no  positive 
evidence  of  an  atmosphere  has  ever  been  obtained,  and 
that  if  one  exists  it  is  certainly  several  hundred  times  rarer 
than  the  atmosphere  of  our  earth.  The  most  delicate 
method  of  detecting  such  an  appendage  would  be  by  its 
refracting  the  light  of  a  star  seen  through  it.  As  the 
moon  advances  in  her  monthly  course  around  the  earth,  she 
frequently  appears  to  pass  over  bright  stars.  These  phe- 
nomena are  called  occultations.  Just  before  the  limb  of 
the  moon  appears  to  reach  the  star,  the  latter  will  be  seen 
through  the  moon's  atmosphere,  if  there  is  one,  and  will 
be  displaced  in  a  direction  from  the  moon's  centre.  But 
the  most  careful  observations  have  failed  to  show  the 
slightest  evidence  of  any  such  displacement.  Hence  the 
most  delicate  test  for  a  lunar  atmosphere  gives  no  evi- 
dence whatever  that  it  exists. 

The  spectra  of  stars  when  about  to  be  occulted  have 
also  been  examined  in  order  to  see  whether  any  absorption 
lines  which  might  be  produced  by  the  lunar  atmosphere 
became  visible.  The  evidence  in  this  direction  has  also 
been  negative.  Moreover,  the  spectrum  of  the  moon  itself 
does  not  seem  to  differ  in  the  slightest  from  that  of  the 
sun.  We  conclude  therefore  that  if  there  is  a  lunar  at- 
mosphere, it  is  too  rare  to  exert  any  sensible  absorption 
upon  the  rays  of  light. 

Light  and  Heat  of  the  Moon — Many  attempts  have 
been  made  to  measure  the  ratio  of  the  light  of  the  full 
moon  and  that  of  the  sun.  The  results  have  been  very 
discordant,  but  all  have  agreed  in  showing  that  the  sun 
emits  several  hundred  thousand  times  as  much  light  as  the 
full  moonf  The  last  and  most  careful  determination  is 


332  ASTRONOMY. 

that  of  ZOLLNER,  who  finds  the  sun  to  be  618,000  times  as 
bright  as  the  full  moon. 

The  moon  must  reflect  the  heat  as  well  as  the  light  of 
the  sun,  and  must  also  radiate  a  small  amount  of  its  own 
heat.  But  the  quantities  thus  reflected  and  radiated  are  so 
minute  that  they  have  defied  detection  except  with  the 
most  delicate  instruments  of  research  now  known.  By  col- 
lecting the  moon's  rays  in  the  focus  of  one  of  his  large  re- 
flecting telescopes,  Lord  ROSSE  was  able  to  show  that  a 
certain  amount  of  heat  is  actually  received  from  the 
moon,  and  that  this  amount  varies  with  the  moon's  phase, 
as  it  should  do.  He  also  sought  to  learn  how  much  of 
the  moon's  heat  was  reflected  and  how  much  radiated. 
This  he  did  by  ascertaining  its  capacity  for  passing 
through  glass.  It  is  well  known  to  students  of  physics 
that  a  very  much  larger  portion  of  the  heat  radiated  by 
the  sun  or  other  extremely  hot  bodies  will  pass  through 
glass  than  of  heat  radiated  by  a  cooler  body.  Experiments 
show  that  about  86  per  cent  of  the  sun's  heat  will  pass 
through  ordinary  optical  glass.  If  the  heat  of  the  moon 
were  entirely  reflected  sun  heat,  it  would  possess  the  same 
property,  and  the  same  proportion  would  pass  through 
glass.  But  the  experiments  of  Lord  ROSSE  have  shown 
that  instead  of  86  per  cent,  only  1 2  per  cent  passed  through 
the  glass.  As  a  general  result  of  all  his  researches,  it  may 
be  supposed  that  about  six  sevenths  of  the  heat  given  out 
by  the  moon  is  radiated  and  one  seventh  reflected. 

Is  there  any  change  on  the  surface  of  the  Moon? — 
When  the  surface  of  the  moon  was  first  found  to  be  cov- 
ered by  craters  having  the  appearance  of  volcanoes  at  the 
surface  of  the  earth,  it  was  very  naturally  thought  that 
these  supposed  volcanoes  might  be  still  in  activity,  and  ex- 
hibit themselves  to  our  telescopes  by  their  flames.  Sir 
WILLIAM  HERSCHEL  supposed  that  he  saw  several  such  vol- 
canoes, and,  on  his  authority,  they  were  long  believed  to 
exist.  Subsequent  observations  have  shown  that  this  was 
a  mistaken  opinion,  though  a  very  natural  one  under  the 


CHANGES  ON  THE  MOON.  333 

circumstances.  If  we  look  at  the  moon  with  a  telescope 
when  she  is  three  or  four  days  old,  we  shall  see  the  darker 
portion  of  her  surface,  which  is  not  reached  by  the  sun's 
rays,  to  be  faintly  illuminated  by  light  reflected  from  the 
earth.  This  appearance  may  always  be  seen  at  the  right 
time  with  the  naked  eye.  If  the  telescope  has  an  aperture 
of  five  inches  or  upward,  and  the  magnifying  power  does 
not  exceed  ten  to  the  inch,  we  shall  generally  see  one  or 
more  spots  on  this  dark  hemisphere  of  the  moon  so  much 
brighter  than  the  rest  of  the  surface  that  they  may  well 
suggest  the  idea  of  being  self-luminous.  It  is,  however, 
known  that  these  are  only  spots  possessing  the  power  of 
reflecting  back  an  unusually  large  portion  of  the  earth's 
light.  Not  the  slightest  sound  evidence  of  any  incandes- 
cent eruption  at  the  moon's  surface  has  ever  been  found. 

Several  instances  of  supposed  changes  on  the  moon's 
surface  have  been  described  in  recent  times.  A  few  years 
ago  a  spot  known  as  Linnaeus,  near  the  centre  of  the 
moon's  visible  disk,  was  found  to  present  an  appearance 
entirely  different  from  its  representation  on  the  map  of 
BEER  and  MAEDLER,  made  forty  years  before.  More 
recently  KLEIN,  of  Cologne,  supposed  himself  to  have  dis- 
covered a  yet  more  decided  change  in  another  feature  of 
the  moon's  surface. 

The  question  whether  these  changes  are  proven  is  one 
on  which  the  opinions  of  astronomers  differ.  The  difficul- 
ty of  reaching  a  certain  conclusion  arises  from  the  fact  that 
each  feature  necessarily  varies  in  appearance,  owing  to  the 
different  ways  in  which  the  sun's  light  falls  upon  it. 
Sometimes  the  changes  are  very  difficult  to  account  for, 
even  when  it  is  certain  that  they  do  not  arise  from  any 
change  on  the  moon  itself.  Hence  while  some  regard  the 
apparent  changes  as  real,  others  regard  them  as  due  only 
to  differences  in  the  mode  of  illumination. 


CHAPTER  V. 

THE   PLANET   MARS. 
§   1.    DESCRIPTION  OF  THE  PLANET. 

Mars  is  the  next  planet  beyond  the  earth  in  the  order 
of  distance  from  the  sun,  being  about  half  as  far  again  as 
the  earth.  It  has  a  decided  red  color,  by  which  it  may 
be  readily  distinguished  from  all  the  other  planets. 
Owing  to  the  considerable  eccentricity  of  its  orbit,  its 
distance,  both  from  the  sun  and  from  the  earth,  varies  in  a 
larger  proportion  than  does  that  of  the  other  outer  planets. 

At  the  most  favorable  oppositions,  its  distance  from  the 
earth  is  about  0-38  of  the  astronomical  unit,  or,  in  round 
numbers,  57,000,000  kilometres  (35,000,000  of  miles). 
This  is  greater  than  the  least  distance  of  Venus,  but  we 
can  nevertheless  obtain  a  better  view  of  Mars  under  these 
circumstances  than  of  Venus,  because  when  the  latter  is 
nearest  to  us  its  dark  hemisphere  is  turned  toward  us, 
while  in  the  case  of  Mars  and  of  the  outer  planets  the 
hemisphere  turned  toward  us  at  opposition  is  fully  illu- 
minated by  the  sun. 

The  period  of  revolution  of  Mars  around  the  sun  is  a 
little  less  than  two  years,  or,  more  exactly,  687  days.  The 
successive  oppositions  occur  at  intervals  of  two  years  and 
one  or  two  months,  the  earth  having  made  during  this 
interval  a  little  more  than  two  revolutions  around  the  sun, 
and  the  planet  Mars  a  little  more  than  one.  The  dates 
of  several  past  and  future  oppositions  are  shown  in  the 
following  table  : 


OPPOSITIONS  OF  MARS.  335 

1871 March  20th. 

1873 April  27th. 

1875 June  20th. 

1877 September  5th. 

1879 November  12th. 

1881 December  26th. 

1884 January  31st. 

1886 March  6th. 

Owing  to  the  unequal  motion  of  the  planet,  arising  from 
the  eccentricity  of  its  orbit,  the  intervals  between  suc- 
cessive oppositions  vary  from  two  years  and  one  month  to 
two  years  and  two  and  a  half  months. 

About  August  26th  of  each  year  the  earth  is  in  the  same 
direction  from  the  sun  as  the  perihelion  of  the  orbit  of 
Mars.  Hence  if  an  opposition  occurs  about  that  time, 
Mars  will  be  very  near  its  perihelion,  and  at  the  least 
possible  distance  from  the  earth.  At  the  opposite  season 
of  the  year,  near  the  end  of  February,  the  earth  is  on 
the  line  drawn  from  the  sun  to  the  aphelion  of  the  orbit 
Mars.  The  least  favorable  oppositions  are  therefore 
those  which  occur  in  February.  The  distance  of  Mars  is 
then  about  0-65  of  the  astronomical  unit. 

The  favorable  oppositions  occur  at  intervals  of  15  or 
17  years,  the  period  being  that  required  for  the  successive 
increments  of  one  or  two  months  between  the  times  of  the 
year  at  which  successive  oppositions  occur  to  make  up  an 
entire  year.  This  will  be  readily  seen  from  the  preceding 
table  of  the  times  of  opposition,  which  shows  how  the  op- 
positions ranged  through  the  entire  year  between  1871 
and  1886.  The  opposition  of  1877  was  remarkably  fa- 
vorable. The  next  most  favorable  opposition  will  occur 
in  1893. 

Mars  necessarily  exhibits  phases,  but  they  are  not  so 
well  marked  as  in  the  case  of  Venus,  because  the  hemi- 
sphere which  it  presents  to  the  observer  on  the  earth  is 
always  more  than  half  illuminated.  The  greatest  phase 


336  ASTRONOMY. 

occurs  when  its  direction  is  90°  from  that  of  the  sun,  and 
even  then  six  sevenths  of  its  disk  is  illuminated,  like  that 
of  the  moon,  three  days  before  or  after  full  moon.  The 
phases  of  Mars  were  observed  by  GALILEO  in  1610,  who, 
however,  could  not  describe  them  with  entire  certainty. 

Rotation  of  Mars. — The  early  telescopic  observers 
noticed  that  the  disk  of  Mars  did  not  appear  uniform  in 
color  and  brightness,  but  had  a  variegated  aspect.  In 
1666  the  celebrated  Dr.  ROBERT  HOOKE  found  that  the 
markings  on  Mars  were  permanent  and  moved  around  in 
such  a  way  as  to  show  that  the  planet  revolved  on  its  axis. 
The  markings  given  in  his  drawing  can  be  traced  at  the 
present  day,  and  are  made  use  of  to  determine  the  exact 
period  of  rotation  of  the  planet.  Drawings  made  by 
HUYGHENS  about  the  same  time  have  been  used  in  the 
same  way.  So  well  is  the  rotation  fixed  by  them  that  the 
astronomer  can  now  determine  the  exact  number  of  times 
the  planet  has  rotated  on  its  axis  since  these  old  drawings 
were  made.  The  period  has  been  found  by  Mr.  PKOCTOK 
to  be  24h  37m  22s  •  7,  a  result  which  appears  certain  to  one 
or  two  tenths  of  a  second.  It  is  therefore  less  than  an 
hour  greater  than  the  period  of  rotation  of  the  earth. 

Surface  of  Mars. — The  most  interesting  result  of  these 
markings  on  Mars  is  the  probability  that  its  surface  is  di- 
versified by  land  and  water,  covered  by  an  atmosphere, 
and  altogether  very  similar  to  the  surface  of  the  earth. 
Some  portions  of  the  surface  are  of  a  decided  red  color, 
and  thus  give  rise  to  the  well-known  fiery  aspect  of  the 
planet.  Other  parts  are  of  a  greenish  hue,  and  are  there- 
fore supposed  to  be  seas.  The  most  striking  features  are 
two  brilliant  white  regions,  one  lying  around  each  pole  of  the 
planet.  It  has  been  supposed  that  this  appearance  is  due 
to  immense  masses  of  snow  and  ice  surrounding  the  poles. 
If  this  were  so,  it  would  indicate  that  the  processes  of  evap- 
oration, cloud  formation,  and  condensation  of  vapor  into 
rain  and  snow  go  on  at  the  surface  of  Mars  as  at  the  sur- 
face of  the  earth.  A  certain  amount  of  color  is  given  to 


ASPECT  OF  MARS.  337 

this  theory  by  supposed  changes  in  the  magnitude  of 
those  ice-caps.  But  the  problem  of  establishing  such 
changes  is  one  of  extreme  difficulty.  The  only  way  in 
which  an  adequate  idea  of  this  difficulty  can  be  formed  is 
by  the  reader  himself  looking  at  Mars  through  a  telescope. 
If  he  will  then  note  how  hard  it  is  to  make  out  the 
different  shades  of  light  and  darkness  on  the  planet,  and 


FlG.    90. — TELESCOPIC  VIEW  OP  MAKS. 

how  they  must  vary  in  aspect  under  different  conditions 
of  clearness  in  our  own  atmosphere,  he  will  readily  per- 
ceive that  much  evidence  is  necessary  to  establish  great 
changes.  All  we  can  say,  therefore,  is  that  the  formation 
of  the  ice-caps  in  winter  and  their  melting  in  summer  has 
some  evidence  in  its  favor,  but  is  not  yet  completely 
proven. 


338  ASTRONOMY. 

§  2.    SATELLITES  OP  MARS. 

Until  the  year  1877,  Mars  was  supposed  to  have  no  sat- 
ellites, none  having  ever  been  seen  in  the  most  powerful 
telescopes.  But  in  August  of  that  year,  Professor  HALL, 
of  the  IS  aval  Observatory,  instituted  a  systematic  search 
with  the  great  equatorial,  which  resulted  in  the  discovery 
of  two  such  objects.  We  have  already  described  the  op- 
position of  1877  as  an  extremely  favorable  one  ;  otherwise 
it  would  have  been  hardly  possible  to  detect  these  bodies. 
They  had  never  before  been  seen,  partly  on  account  of 
their  extreme  minuteness,  which  rendered  them  invisible 
except  with  powerful  instruments  and  at  the  most  favor- 
able times,  and  partly  on  account  of  the  fact,  already  al- 
luded to,  that  the  favorable  oppositions  occur  only  at  inter- 
vals of  15  or  17  years.  There  are  only  a  few  weeks  dur- 
ing each  of  these  intervals  when  it  is  practicable  to  distin- 
guish them. 

These  satellites  are  by  far  the  smallest  celestial  bodies 
known.  It  is  of  course  impossible  to  measure  their  diam- 
eters, as  they  appear  in  the  telescope  only  as  points  of 
light.  A  very  careful  estimate  of  the  amount  of  light 
which  they  reflect  was  made  by  Professor  E.  C.  PICKER- 
ING, Director  of  the  Harvard  College  Observatory,  who 
calculated  how  large  they  ought  to  be  to  reflect  this  light. 
He  thus  found  that  the  outer  satellite  was  probably  about 
six  miles  and  the  inner  one  about  seven  miles  in  diameter, 
supposing  them  to  reflect  the  solar  rays  precisely  as  Mars 
does.  The  outer  one  was  seen  with  the  telescope  at  a  dis- 
tance from  the  earth  of  7,000,000  times  this  diameter. 
The  proportion  would  be  that  of  a  ball  two  inches  in  di- 
ameter viewed  at  a  distance  equal  to  that  between  the 
cities  of  Boston  and  New  York.  Such  a  feat  of  telescopic 
seeing  is  well  fitted  to  give  an  idea  of  the  power  of  modern 
optical  instruments. 

Professor  HALL  found  that  the  outer  satellite,  which 
he  called  Deimos,  revolves  around  the  planet  in  30h  10m, 


SATELLITES  OF  MARS.  339 

and  the  inner  one,  called  Pholjos,  in  7h  38m.  Tlie  latter  is 
only  5800  miles  from  the  centre  of  Mars,  and  less  than 
4000  miles  from  its  surface.  It  would  therefore  be  almost 
possible  with  one  of  our  telescopes  on  the  surface  of  Mars 
to  see  an  object  the  size  of  a  large  animal  on  the  satellite. 
This  short  distance  and  rapid  revolution  make  the  inner 
satellite  of  Mars  one  of  the  most  interesting  bodies  with 
which  we  are  acquainted.  It  performs  a  revolution  in  its 
orbit  in  less  than  half  the  time  that  Mars  revolves  on  its 
axis.  In  consequence,  to  the  inhabitants  of  Mars,  it 
would  seem  to  rise  in  the  west  and  set  in  the  east.  It  will 
be  remembered  that  the  revolution  of  the  moon  around 
the  earth  and  of  the  earth  on  its  axis  are  both  from  west 
to  east ;  but  the  latter  revolution  being  the  more  rapid,  the 
apparent  diurnal  motion  of  the  moon  is  from  east  to  west. 
In  the  case  of  the  inner  satellite  of  Mars,  however,  this 
is  reversed,  and  it  therefore  appears  to  move  in  the  actual 
direction  of  its  orbital  motion.  The  rapidity  of  its  phases 
is  also  equally  remarkable.  It  is  less  than  two  hours  from 
new  moon  to  first  quarter,  and  so  on.  Thus  the  inhabit- 
ants of  Mars  may  see  their  inner  moon  pass  through  all 
its  phases  in  a  single  night. 


CHAPTER  VI. 

THE   MINOR  PLANETS. 

WHEN  the  solar  system  was  first  mapped  out  in  its  true 
proportions  by  COPERNICUS  and  KEPLER,  only  six  primary 
planets  were  known  —  namely,  Mercury r,  Venus,  the 
Earth,  Mars,  Jupiter,  and  Saturn.  These  succeeded 
each  other  according  to  a  nearly  regular  law,  as  we  have 
shown  in  Chapter  I. ,  except  that  between  Mars  and  Jupi- 
ter a  gap  was  left,  where  an  additional  planet  might  be 
inserted,  and  the  order  of  distance  be  thus  made  complete. 
It  was  therefore  supposed  by  the  astronomers  of  the  seven- 
teenth and  eighteenth  centuries  that  a  planet  might  be 
found  in  this  region.  A  search  for  this  object  was  insti- 
tuted toward  the  end  of  the  last  century,  but  before  it 
had  made  much  progress  a  planet  in  the  place  of  the  one 
so  long  expected  was  found  by  PIAZZI,  of  Palermo.  The 
discovery  was  made  on  the  first  day  of  the  present  century, 
1801,  January  1st. 

In  the  course  of  the  following  seven  years  the  astronom- 
ical world  was  surprised  by  the  discovery  of  three  other 
planets,  all  in  the  same  region,  though  not  revolving  in 
the  same  orbits.  Seeing  four  small  planets  where  one 
large  one  ought  to  be,  OLBERS  was  led  to  his  celebrated 
hypothesis  that  these  bodies  were  the  fragments  of  a  large 
planet  which  had  been  broken  to  pieces  by  the  action  of 
some  unknown  force. 

A  generation  of  astronomers  now  passed  away  without 
the  discovery  of  more  than  these  four.  But  in  December, 
1845,  HKNOKTC,  of  Dreisen,  being  engaged  in  mapping 


THE  MINOR  PLANETS.  341 

down  the  stars  near  the  ecliptic,  found  a  fifth  planet  of 
the  group.  In  1847  three  more  were  discovered,  and 
discoveries  have  since  been  made  at  a  rate  which  thus  far 
shows  no  signs  of  diminution.  The  number  has  now 
reached  200,  and  the  discovery  of  additional  ones  seems  to 
be  going  on  as  fast  as  ever.  The  frequent  announcements 
of  the  discovery  of  planets  which  appear  in  the  public 
prints  all  refer  to  bodies  of  this  group. 

The  minor  planets  are  distinguished  from  the  major 
ones  by  many  characteristics.  Among  these  we  may 
mention  their  great  number,  which  exceeds  that  of  all  the 
other  known  bodies  of  the  solar  system  ;  their  small  size  ; 
their  positions,  all  being  situated  between  the  orbits  of 
Mars  and  Jupiter  y  the  great  eccentricities  and  inclina- 
tions of  their  orbits. 

Number  of  Small  Planets. — It  would  be  interesting  to 
know  how  many  of  these  planets  there  are  in  all,  but  it  is 
as  yet  impossible  even  to  guess  at  the  number.  As 
already  stated,  fully  200  are  now  known,  and  the  number 
of  new  ones  found  every  year  ranges  from  7  or  8  to  10  or 
12.  If  ten  additional  ones  are  found  every  year  during 
the  remainder  of  the  century,  400  will  then  have  been 
discovered. 

The  discovery  of  these  bodies  is  a  very  difficult  work, 
requiring  great  practice  and  skill  on  the  part  of  the  as- 
tronomer. The  difficulty  is  that  of  distinguishing  them 
amongst  the  hundreds  of  thousands  of  telescopic  stars 
which  are  scattered  in  the  heavens.  A  minor  planet 
presents  no  sensible  disk,  and  therefore  looks  exactly  like 
a  small  star.  It  can  be  detected  only  by  its  motion  among 
the  surrounding  stars,  which  is  so  slow  that  hours  or  even 
days  must  elapse  before  it  can  be  noticed. 

Magnitudes. — In  consequence  of  the  minor  planets  hav- 
ing no  visible  disks  in  the  most  powerful  telescopes,  it  is  im- 
possible to  make  any  precise  measurement  of  their  diam- 
eters. These  can,  however,  be  estimated  by  the  amount 
of  light  which  the  planet  reflects.  Supposing  the  propor- 


342  ASTRONOMY. 

tion  of  light  reflected  about  the  same  as  in  the  case  of  the 
larger  planets,  it  is  estimated  that  the  diameters  of  the 
three  or  four  largest,  which  are  those  first  discovered, 
range  between  300  and  600  kilometres,  while  the  smallest 
are  probably  from  20  to  50  kilometres  in  diameter.  The 
average  diameter  of  all  that  are  known  is  perhaps  less  than 
150  kilometres — that  is,  scarcely  more  than  one  hundredth 
that  of  the  earth.  The  volumes  of  solid  bodies  vary  as  the 
cubes  of  their  diameters  ;  it  might  therefore  take  a  million 
of  these  planets  to  make  one  of  the  size  of  the  earth. 

Form  of  Orbits.— The  orbits  of  the  minor  planets  are  much 
more  eccentric  than  those  of  the  larger  ones  ;  their  distance  from 
the  sun  therefore  varies  very  widely.  The  most  eccentric  orbit  yet 
known  is  that  of  Aethra,  which  was  discovered  by  Professor  WAT- 
SON in  1873.  Its  least  distance  from  the  sun  is  1*61,  a  very  little 
further  than  Mars,  while  at  aphelion  it  is  3 -59,  or  more  than  twice 
as  far.  Two  or  three  others  are  twice  as  far  from  the  sun  at  aphe- 
lion as  at  perihelion,  while  nearly  all  are  so  eccentric  that  if  the 
orbits  were  drawn  to  a  scale,  the  eye  would  readily  perceive  that  the 
sun  was  not  in  their  centres.  The  largest  inclination  of  all  is  that 
of  Pallas,  which  is  one  of  the  original  four,  having  been  discovered 
by  OLBERS  in  1802.  The  inclination  to  the  ecliptic  is  34°,  or  more 
than  one  third  of  a  right  angle.  Five  or  six  others  have  inclinations 
exceeding  20°;  they  therefore  range  entirely  outside  the  zodiac,  and 
in  fact  sometimes  culminate  to  the  north  of  our  zenith. 

Origin  of  the  Minor  Planets. — The  question  of  the  origin  of 
these  bodies  was  long  one  of  great  interest.  The  features  which  we 
have  described  associate  themselves  very  naturally  with  the  cele- 
brated hypothesis  of  OLBERS,  that  we  here  have  the  fragments  of  a 
single  large  planet  which  in  the  beginning  revolved  in  its  proper 
place  between  the  orbits  of  Mars  and  Jupiter.  OLBERS  himself  sug- 
gested a  test  of  his  theory.  If  these  bodies  were  really  formed  by 
an  explosion  of  the  large  one,  the  separate  orbits  of  the  fragments 
would  all  pass  through  the  point  where  the  explosion  occurred.  A 
common  point  of  intersection  was  therefore  long  looked  for  ;  but 
although  two  or  three  of  the  first  four  did  pass  pretty  near  each 
other,  the  required  point  could  not  be  found  for  all  four. 

It  was  then  suggested  that  the  secular  changes  in  the  orbits  pro- 
duced by  the  action  of  the  other  planets  would  in  time  change  the 
positions  of  all  the  orbits  in  such  a  way  that  they  would  no  longer 
have  any  common  intersection.  The  secular  variations  of  their  orbits 
were  therefore  computed,  to  see  if  there  was  any  sign  of  the  required 
intersection  in  past  ages,  but  none  could  be  found.  No  support 
has  been  given  to  OLBERS?  hypothesis  by  subsequent  investigations, 
and  it  is  no  longer  considered  by  astronomers  to  have  any  founda- 
tion. So  far  as  can  be  judged,  these  bodies  have  been  revolving 
around  the  sun  as  separate  planets  ever  since  the  solar  system  itself 
was  formed. 


CHAPTER  VII. 

JUPITER  AND   HIS   SATELLITES. 
§   1.    THE  PLANET  JUPITER. 

Jupiter  is  much  the  largest  planet  in  the  system.  His 
mean  distance  is  nearly  800,000,000  kilometres  (480,000,- 
000  miles).  His  diameter  is  140,000  kilometres,  corre- 
sponding to  a  mean  apparent  diameter,  as  seen  from  the 
sun  of  36" -5.  His  linear  diameter  is  about  y^,  his  surface 
is  T-Jo,  and  his  volume  T¥Vo- tnat  °^  tne  sun-  His  mass  is 
-j-J^-,  and  his  density  is  thus  nearly  the  same  as  the  sun's — 
viz.  ,0-24  of  the  earth's.  He  rotates  on  his  axis  in  9h  55m  20s. 

He  is  attended  by  four  satellites,  which  were  discovered 
by  GALILEO  on  January  7th,  1610.  He  named  them  in 
honor  of  the  MEDICIS,  the  Medicean  stars.  These  satellites 
were  independently  discovered  on  January  16th,  1610,  by 
HARRIOT,  of  England,  who  observed  them  through  several 
subsequent  years.  SIMON  MARIUS  also  appears  to  have 
early  observed  them,  and  the  honor  of  their  discovery  is 
claimed  for  him.  They  are  now  known  as  Satellites  I, 
II,  III,  and  IY,  I  being  the  nearest. 

The  surface  of  Jupiter  has  been  carefully  studied  with 
the  telescope,  particularly  within  the  past  20  years.  Al- 
though further  from  us  than  Mars,  the  details  of  his  disk 
are  much  easier  to  recognize.  The  most  characteristic 
features  are  given  in  the  drawings  appended.  These  feat- 
ures are,  firstly,  the  dark  bands  of  the  equatorial  regions, 
arid,  secondly,  the  cloud-like  forms  spread  over  nearly  the 
whole  surface.  At  the  limb  all  these  details  become  indis- 


344  ASTRONOMY. 

tinct,  and  finally  vanish,  thus  indicating  a  highly  absorptive 
atmosphere.  The  light  from  the  centre  of  the  disk  is  twice 
as  bright  as  that  from  the  poles  (AKAGO).  The  bands  can 
be  seen  with  instruments  no  more  powerful  than  those 
used  by  GALILEO,  yet  he  makes  no  mention  of  them,  al- 
though they  were  seen  by  ZUCCHI,  FONT  ANA,  and  others  be- 
fore 1633.  HUYGHENS  (1659)  describes  the  bands  as 
brighter  than  the  rest  of  the  disk — a  unique  observation, 
on  which  we  must  look  with  some  distrust,  as  since  1660 
they  have  constantly  been  seen  darker  than  the  rest  of  the 
planet. 

The  color  of  the  bands  is  frequently  described  as  a  brick- 
red,  but  one  of  the  authors  has  made  careful  studies  in 


FlG.    91.— TELESCOPIC  VIEW  OP  JUPITER  AND  HIS  SATELLITES. 

color  of  tliis  planet,  and  finds  the  prevailing  tint  to  be  a 
salmon  color,  exactly  similar  to  the  color  of  Mars.  The 
position  of  the  bands  varies  in  latitude,  and  the  shapes  of 
the  limiting  curves  also  change  from  day  to  day  ;  but  in 
the  main  they  remain  as  permanent  features  of  the  region 
to  which  they  belong.  Two  such  bands  are  usually  vis- 
ible, but  often  more  are  seen.  For  example,  CASSINI 
(1690,  December  16th)  saw  six  parallel  bands  extending 
completely  around  the  planet.  HERSCIIEL,  in  the  year 
1793,  attributed  the  aspects  of  the  bands  to  zones  of  the 
planet's  atmosphere  more  tranquil  and  less  filled  with 
clouds  than  the  remaining  portions,  so  as  to  permit  the 


ASPECT  OF  JUPITER.  345 

true  surface  of  the  planet  to  be  seen  through  these  zones, 
while  the  prevailing  clouds  in  the  other  regions  give 
a  brighter  tint  to  these  latter.  The  color  of  the  bands 
seems  to  vary  from  time  to  time,  and  their  bordering 
lines  sometimes  alter  with  such  rapidity  as  to  show  that 
these  borders  are  formed  of  something  like  clouds. 

The  clouds  themselves  can  easily  be  seen  at  times,  and 
they  have  every  variety  of  shape,  sometimes  appearing  as 


FlG.    92. — TELESCOPIC    VIEW    OP    JUPITER,     WITH    A    SATELLITE    AND 
ITS  SHADOW  SEEN  ON  IT. 

brilliant  circular  white  masses,  but  oftener  they  are  similar 
in  form  to  a  series  of  white  cumulous  clouds  such  as  are 
frequently  seen  piled  up  near  the  horizon  on  a  summer's 
day.  The  bands  themselves  seem  frequently  to  be  veiled 
over  with  something  like  the  thin  cirrus  clouds  of  our 
atmosphere.  On  one  occasion  an  annulus  of  white  cloud 
was  seen  on  one  of  the  dark  bands  for  many  days,  retain- 
ing its  shape  through  the  whole  period. 


346  ASTRONOMY. 

Such  clouds  can  be  tolerably  accurately  observed,  and 
may  be  used  to  determine  $ie  rotation  time  of  the  planet. 
These  observations  show  that  the  clouds  have  often  a 
motion  of  their  own,  which  is  also  evident  from  other  con- 
siderations. 

The  following  results  of  observation,  of  spots  situated  in 
various  regions  of  the  planet  will  illustrate  this  : 

h.  m.  s. 

CASSINI 1665,  rotation  time  =  9  56  00 

HERSCHEL 1778,  '      =  9  55  40 

HERSCHEL 1779,  '            '      =  9  50  48 

SCHROETER 1785,  '            '      =  9  56  56 

BEER  &MADLER.  ...  1835,  '            '      =  9  55  26 

AIRY :....  1835,  «            '      =  9  55  21 

SCHMIDT 1862,  '            •      =  9  55  29 


§  2.  THE  SATELLITES  OP  JUPITER. 

Motions  of  the  Satellites. —  The  four  satellites  move 
about  Jupiter  from  west  to  east  in  nearly  circular  orbits. 
When  one  of  these  satellites  passes  between  the  sun  and 
Jupiter,  it  casts  a  shadow  upon  Jupiter  *s  disk  (see  Fig.  92) 
precisely  as  the  shadow  of  our  moon  is  thrown  upon  the 
earth  in  a  solar  eclipse.  If  the  satellite  passes  through 
Jupiter'' s  own  shadow  in  its  revolution,  an  eclipse  of  this 
satellite  takes  place.  If  it  passes  between  the  earth  and 
Jupiter,  it  is  projected  upon  Jupiter 's  disk,  and  we  have  a 
transit  ;  if  Jupiter  is  between  the  earth  and  the  satellite, 
an  occultation  of  the  latter  occurs.  All  these  phenomena 
can  be  seen  from  the  earth  with  a  common  telescope,  and 
the  times  of  observation  are  all  found  predicted  in  the 
Nautical  Almanac.  In  this  way  we  are  sure  that  the  black 
spots  which  we  see  moving  across  the  disk  of  Jupiter  are 
really  the  shadows  of  the  satellites  themselves,  and  not  phe- 
nomena to  be  otherwise  explained.  These  shadows  being 
seen  black  upon  Jupiter* s  surface,  show  that  this  planet 
shines  by  reflecting  the  light  of  the  sun. 


SATELLITES  OF  JUPITER.  347 

Telescopic  Appearance  of  the  Satellites. — Under  ordi- 
nary circumstances,  the  satellites  of  Jupiter  are  seen  to 
have  disks— that  is,  not  to  be  mere  points  of  light.  Un- 
der very  favorable  conditions,  markings  have  beeen  seen 
on  these  disks,  and  it  is  very  curious  that  the  anomalous 
appearances  given  in  Fig.  93  (by  Dr.  HASTINGS)  have  been 
seen  at  various  times  by  other  good  observers,  as  SECCHI, 
DA  WES,  and  RUTHEKFUKD.  Satellite  III,  which  is  much 
the  largest,  has  decided  markings  on  its  face  ;  IY  some- 
times appears,  as  in  the  figure,  to  have  its  circular  outline 


FlG.  93. — TELESCOPIC  APPEARANCE  OF  JUPITER'S  SATELLITES. 

cut  off  by  right  lines,  and  satellite  I  sometimes  appears 
gibbous.  The  opportunities  for  observing  these  appear- 
ances are  so  rare  that  nothing  is  known  beyond  the  bare 
fact  of  their  existence,  and  no  plausible  explanation  of  the 
figure  shown  in  IV  has  been  given. 

Phenomena  of  the  Satellites. — The  phenomena  of  the  satel- 
lites are  illustrated  in  Fig.  94.  Here  8  represents  the  sun,  A  T 
the  orbit  of  the  earth  (the  earth  itself  being  at  T),  the  outer  circle 
the  orbit  of  Jupiter,  and  the  four  small  circles  upon  the  latter  four 
different  positions  of  the  orbit  of  a  satellite.  In  the  centre  of  each 
of  the  satellite  orbits  will  be  seen  a  small  white  circle  designed  to 
represent  the  planet  Jupiter  itself.  The  dotted  lines  drawn  from 
each  edge  of  the  sun  to  the  corresponding  edges  of  the  planet  and 
continued  until  they  meet  in  a  point  show  the  outlines  of  the 
shadow  of  Jupiter. 

Let  us  first  consider  the  position  of  Jupiter  marked  <7to  the  left 
of  the  figure,  it  being  then  in  opposition  to  the  sun.  The  observer 
on  the  earth  at  T  could  not  then  see  an  object  anywhere  in  the 
shadow  of  Jupiter  because  the  latter  is  entirely  behind  the  planet. 
Hence,  as  the  satellite  moves  around,  he  will  see  it  disappear  behind 
the  right-hand  limb  of  the  planet  and  reappear  from  the  left-hand 
limb.  Such  a  phenomenon  is  called  an  occultation,  and  is  desig- 
nated as  disappearance  or  reappearance,  according  to  the  phase. 

It  may  be  remarked,  however,  that  the  inclination  of  the  outer 
satellite  to  the  orbit  of  Jupiter  is  so  great  that  it  sometimes  passes 


348  ASTRONOMY. 

entirely  above  or  below  the  planet,  and  therefore  is  not  occulted 
at  all. 

Let  us  next  consider  Jupiter  in,  the  position  J"  near  the  bottom  of 
the  figure,  the  shadow,  as  before,  pointing  from  the  planet  directly 
away  from  the  sun.  If  the  shadow  were  a  visible  object,  the  ob- 
server on  the  earth  at  T  could  see  it  projected  out  on  the  right  of 
the  planet,  because  he  is  not  in  the  line  between  Jupiter  and  the  sun. 
Hence  as  a  satellite  moves  around  and  enters  the  shadow,  he  will  see 
it  disappear  from  sight,  owing  to  the  sunlight  being  cut  off ;  this 


FlG.  94.— PHENOMENA  OF  JUPITER'S  SATELLITES. 

is  called  an  eclipse  disappearance.  If  the  satellite  is  one  of  the  two 
outer  ones,  he  will  be  able  to  see  it  reappear  again  after  it  comes 
out  of  the  shadow  before  it  is  occulted  behind  the  planet. 

Soon  afterward  the  occupation  will  occur,  and  it  will  afterward 
reappear  on  the  left.  In  the  case  of  the  inner  or  first  satellite,  how- 
ever, the  point  of  emergence  from  the  shadow  is  hidden  behind  the 
planet,  consequently  the  observer,  after  it  once  disappears  in  the  shad- 
ow, will  not  see  it  reappear  until  it  emerges  from  behind  the  planet. 

If  the  planet  is  in  the   position  J1,  the  satellite  will   be   occulted 


SATELLITES  OF  JUPITER.  349 

behind  the  planet  where  it  reaches  the  first  dotted  line.  If  it  is  the  in- 
ner satellite,  it  will  not  be  seen  to  reappear  on  the  other  side  of  the 
planet,  because  when  it  reaches  the  second  dotted  line  it  has  entered 
the  shadow.  After  a  while,  however,  it  will  reappear  from  the 
shadow  some  little  distance  to  the  left  of  the  planet  ;  this  phe- 
nomenon is  called  an  eclipse  reappearance.  In  the  case  of  the  outer 
satellites,  it  may  sometimes  happen  that  they  are  visible  for  a  short 
time  after  they  emerge  from  behind  the  disk  and  before  they  enter 
the  shadow. 

These  different  appearances  are,  for  convenience,  represented  in 
the  figure  as  corresponding  to  different  positions  of  Jupiter  in  his 
orbit,  the  earth  having  the  same  position  in  all ;  but  since  Jupiter 
revolves  around  the  sun  only  once  in  twelve  years,  the  changes  of 
relative  position  really  correspond  to  different  positions  of  the  earth 
in  its  orbit  during  the  course  of  the  year. 

The  satellites  completely  disappear  from  telescopic  view  when 
they  enter  the  shadow  of  the  planet.  This  seems  to  show  that 
neither  planet  nor  satellite  is  self-luminous  to  any  great  extent.  If  the 
satellite  were  self-luminous,  it  would  be  seen  by  its  own  light,  and 
if  the  planet  were  luminous  the  satellite  might  be  seen  by  the  re- 
flected light  of  the  planet. 

The  motions  of  these  objects  are  connected  by  two  curious  and 
important  relations  discovered  by  LA  PLACE,  and  expressed  as  fol- 
lows: 

I.  The  mean  motion  of  the  first  satellite  added  to  twice  the,    mean 
motion  of  the  third  is  exactly  equal  to  three  times  the  mean  motion  of 
the  second. 

II.  If  to  the  mean  longitude  of  the  first  satellite  we  add  twice  the 
mean  longitude  of  the  third,  and  subtract  three  times  the  mean  longitude 
of  the  second,  the  difference  is  always  180°. 

The  first  of  these  relations  is  shown  in  the  following  table  of  the 
mean  daily  motions  of  the  satellites: 

Satellite   I  in  one  day  moves 203°  •  4890 

II        "          "            101°. 3748 

•'      III        "          "            50°-3177 

IV        "          "            21C- 5711 

Motion  of  Satellite  1 203° -4890 

Twice  that  of  Satellite  III 100°  •  6354 

Sum 304°  -1244 

Three  times  motion  of  Satellite  II 304°  -1244 

Observations  showed  that  this  condition  was  fulfilled  as  exactly 
as  possible,  but  the  discovery  of  LA  PLACE  consisted  in  showing  that 
if  the  approximate  coincidence  of  the  mean  motions  was  once  es- 
tablished, they  could  never  deviate  from  exact  coincidence  with 
the  law.  The  case  is  analogous  to  that  of  the  moon,  which  always 
presents  the  same  face  to  us  and  which  always  will  since  the  rela- 
tion being  once  approximately  true,  it  will  become  t'xact  and  ever 
remain  so. 


350 


ASTRONOMY. 


The  discovery  of  the  gradual  propagation  of  light  by  means  of 
these  satellites  has  already  been  described,  and  it  has  also  been  ex- 
plained that  they  are  of  use  in«the  rough  determination  of  longi- 
tudes. To  facilitate  their  observation,  the  Nautical  Almanac  gives 
complete  ephemerides  of  their  phenomena.  A  specimen  of  a  por- 
tion of  such  an  ephemeris  for  1865,  March  7th,  8th,  and  9th,  is 
added.  The  times  are  Washington  mean  times.  The  letter  W  in- 
dicates that  the  phenomenon  is  visible  in  Washington. 

1865— MARCH. 


d.      h.     m.        s 

I. 

Eclipse 

Disapp. 

7    18    27    38-5 

I. 

Occult. 

Reapp. 

7    21    56 

III. 

Shadow 

Ingress 

8      7    27 

III. 

Shadow 

Egress 

8      9    58 

III. 

Transit 

Ingress 

8    12    31 

II. 

Eclipse 

Disapp. 

8    13      1    22-7 

III. 

Transit 

Egress     W. 

8    15      6 

II. 

Eclipse 

Reapp.     W. 

8    15    24    11-1 

II. 

Occult. 

Disapp.    W. 

8    15    27 

I. 

Shadow 

Ingress    W. 

8    15    43 

( 

Transit 

Ingress    W. 

8    16    58 

Shadow 

Egress 

8    17    57 

I  '. 

Occult. 

Reapp. 

8    17    59 

Transit 

Egress 

8    19    13 

Eclipse 

Disapp. 

9    12    55    59-4 

Occult. 

Reapp.     W. 

9    16    25 

Suppose  an  observer  near  New  York  City  to  have  determined  his 
local  time  accurately.  This  is  about  13m  faster  than  Washington 
time.  On  1865,  March  8th,  he  would  look  for  the  reappearance  of 
II  at  about  15h  34m  of  his  local  time.  Suppose  he  observed  it 
at  15h  36m  22s -7  of  his  time  :  then  his  meridian  is  12m  11s -6 
east  of  Washington.  The  difficulty  of  observing  these  eclipses  with 
accuracy,  and  the  fact  that  the  aperture  of  the  telescope  employed 
has  an  important  effect  on  the  appearances  seen,  have  kept  this 
method  from  a  wide  utility,  which  it  at  first  seemed  to  promise. 

The  apparent  diameters  of  these  satellites  have  been  measured  by 
STKUVE,  SECCHI,  and  others,  and  the  best  results  are  : 

I,  1"-0;  II,  0"-9;  III,  1"'5;  IV,  1"'3. 

Their  masses  (Jupiter =1)  are  : 

I,  0-000017;  II,  0000023;  III,  0-000088;  IV,  0-000043. 

The  third  satellite  is  thus  the  largest,  and  it  has  about  the  den- 
sity of  the  planet.  The  true  diameters  vary  from  2200  to  3700 
miles.  The  volume  of  II  is  about  that  of  our  moon  ;  III  approaches 
our  earth  in  size. 

Variations  in  the  light  of  these  bodies  have  constantly  been 
noticed  which  have  been  supposed  to  be  due  to  the  fact  that  they 
turned  on  their  axes  once  in  a  revolution,  and  thus  presented  various 
faces  to  us.  The  recent  accurate  photometric  measures  of  ENGEL- 
MANN  show  that  this  hypothesis  will  not  account  for  all  the  changes 
observed,  some  of  which  appear  to  be  quite  sudden. 


SATELLITES  OF  JUPITER. 


351 


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CHAPTER  VIII. 

SATURN   AND   ITS    SYSTEM. 
§   1.     GENERAL  DESCRIPTION. 

Saturn  is  the  most  distant  of  tlie  major  planets  known 
to  the  ancients.  It  revolves  around  the  sun  in  29J  years, 
at  a  mean  distance  of  nearly  1,500,000,000  kilometres 
(890,000,000  miles).  The  angular  diameter  of  the  ball  of 
the  planet  is  about  16"  •  2,  corresponding  to  a  true  diam- 
eter of  about  110,000  kilometres  (70,500  miles).  Its  diam- 
eter is  therefore  nearly  nine  times  and  its  volume  about 
700  times  that  of  the  earth.  It  is  remarkable  for  its  small 
density,  which,  so  far  as  known,  is  less  than  that  of  any 
other  heavenly  body,  and  even  less  than  that  of  water. 
Consequently,  it  cannot  be  composed  of  rocks,  like  those 
which  form  our  earth.  It  revolves  on  its  axis,  according 
to  the  recent  observations  of  Professor  HALL,  in  10h  14m 
24s,  or  less  than  half  a  day. 

Saturn  is  perhaps  the  most  remarkable  planet  in  the  so- 
lar system,  being  itself  the  centre  of  a  system  of  its  own, 
altogether  unlike  any  thing  else  in  the  heavens.  Its  most 
noteworthy  feature  is  seen  in  a  pair  of  rings  which  sur- 
round it  at  a  considerable  distance  from  the  planet  itself. 
Outside  of  these  rings  revolve  no  less  than  eight  satellites, 
or  twice  the  greatest  number  known  to  surround  any 
other  planet.  The  planet,  rings,  and  satellites  are  alto- 
gether called  the  Saturnian  system.  The  general  appear- 
ance of  this  system,  as  seen  in  a  small  telescope,  is  shown 
in  Fig.  95. 


I 


ASPECT  OF  SATURN.  353 


To  the  naked  eye,  Saturn  is  of  a  dull  yellowish  color, 
shining  with  about  the  brilliancy  of  a  star  of  the  first  mag- 
nitude. It  varies  in  brightness,  however,  with  the  way 
in  which  its  ring  is  seen,  being  brighter  the  wider  the  ring 
appears.  It  comes  into  opposition  at  intervals  of  one  year 
and  from  twelve  to  fourteen  days.  The  following  are  the 
times  of  some  of  these  oppositions,  by  studying  which  one 
will  be  enabled  to  recognize  the  planet : 


FlG.  95. — TELESCOPIC   VIEW   OF   THE   SATURNIAN   SYSTEM. 

1879 October  5th. 

1880 October  18th. 

1881 October  31st. 

1882 November  14th. 

1883 November  28th. 

1884 December  llth. 

During  these  years  it  will  be  best  seen  in   the  autumn 
and  winter. 


354  ASTRONOMY. 

When  viewed  with  a  telescope,  the  physical  appearance 
of  the  hall  of  Saturn  is  quite  similar  to  that  of  Jupiter ', 
having  light  and  dark  belts  parallel  to  the  direction  of  its 
rotation.  But  these  cloud-like  belts  are  very  difficult  to 
see,  and  so  indistinct  that  it  is  not  easy  to  determine  the 
time  of  rotation  from  them.  This  has  been  done  by  ob- 
serving the  revolution  of  bright  or  dark  spots  which  appear 
on  the  planet  on  very  rare  occasions. 

§   2.    THE  RINGS  OP  SATURN. 

The  rings  are  the  most  remarkable  and  characteristic 
feature  of  the  Saturnian  system.  Fig.  96  gives  two  views 
of  the  ball  and  rings.  The  upper  one  shows  one  of  their 
aspects  as  actually  presented  in  the  telescope,  and  the 
lower  one  shows  what  the  appearance  would  be  if  the 
planet  were  viewed  from  a  direction  at  right  angles  to  the 
plane  of  the  ring  (which  it  never  can  be  from  the  earth). 

The  first  telescopic  observers  of  Saturn  were  unable  to 
see  the  rings  in  their  true  form,  and  were  greatly  per- 
plexed to  account  for  the  appearance  which  the  planet 
presented.  GALILEO  described  the  planet  as  "  tri-corpo- 
rate,"  the  two  ends  of  the  ring  having,  in  his  imperfect 
telescope,  the  appearance  of  a  pair  of  small  planets  at- 
tached to  the  central  one.  ' t  On  each  side  of  old  Saturn 
were  servitors  who  aided  him  on  his  way. ' '  This  sup- 
posed discovery  was  announced  to  his  friend  KEPLER  in 
the  following  logogriph  : 

smaismrmilmepoelalevmibonenogtteviras,  which,  being 
transposed,  becomes — 

"  Altissimam  planetam  tergeminam  obsevavi"  (I  have 
observed  the  most  distant  planet  to  be  triform). 

The  phenomenon  constantly  remained  a  mystery  to  its 
first  observer.  In  1610  he  had  seen  the  planet  accompa- 
nied, as  he  supposed,  by  two  lateral  stars ;  in  1612  the 
latter  had  vanished,  and  the  central  body  alone  remained. 
After  that  GALILEO  ceased  to  observe  $aturn. 


KINGS  OF  SATURN. 


355 


FlG.  96. — RINGS  OP  SATURN. 


356  ASTRONOMY. 

The  appearances  of  the  ring  were  also  incomprehensible 
to  HEVELIUS,  GASSENDI,  and  others.  It  was  not  until 
1655  (after  seven  years  of  observation)  that  the  celebrated 
HUYGHENS  discovered  the  true  explanation  of  the  remark- 
able and  recurring  series  of  phenomena  present  by  the  tri- 
corporate  planet. 

He  announced  his  conclusions  in  the  following  logo- 
griph  :— 

"  aaaaaa  ccccc  d  eeeee  g  h  iiiiiii  1111  mm  nnnnnnimii 
oooo  pp  q  IT  s  ttttt  uuuuu, ' '  which,  when  arranged,  read— 

"  Annulo  cingitur,  tenui,  piano,  nuscjuam  coherente, 
ad  eclipticam  inclinato"  (it  is  girdled  by  a  thin  plane  ring, 
nowhere  touching,  inclined  to  the  ecliptic). 

This  description  is  complete  and  accurate. 

In  1665  it  was  found  by  BALL,  of  England,  that  what 
HUYGHENS  had  seen  as  a  single  ring  was  really  two.  A 
division  extended  all  the  way  around  near  the  outer  edge. 
This  division  is  shown  in  the  figures. 

In  1850  the  Messrs.  BOND,  of  Cambridge,  found  that  there 
was  a  third  ring,  of  a  dusky  and  nebulous  aspect,  inside 
the  other  two,  or  rather  attached  to  the  inner  edge  of  the 
inner  ring.  It  is  therefore  known  as  J3ond^s  dusky  ring. 
It  had  not  been  before  fully  described  owing  to  its  darls:- 
ness  of  color,  which  made  it  a  difficult  object  to  see  except 
with  a  good  telescope.  It  is  not  separated  from  the  bright 
ring,  but  seems  as  if  attached  to  it.  The  latter  shades  off 
toward  its  inner  edge,  which  merges  gradually  into  the 
dusky  ring  so  as  to  make  it  difficult  to  decide  precisely 
where  it  ends  and  the  dusky  ring  begins.  The  latter  ex- 
tends about  one  half  way  from  the  inner  edge  of  the 
bright  ring  to  the  ball  of  the  planet. 

Aspect  of  the  Rings.- — As  Saturn  revolves  around  the 
sun,  the  plane  of  the  rings  remains  parallel  to  itself.  That 
is,  if  we  consider  a  straight  line  passing  through  the  centre 
of  the  planet,  perpendicular  to  the  plane  of  the  ring,  as 
the  axis  of  the  latter,  this  axis  will  always  point  in  the 
same  direction.  In  this  respect,  the  motion  is  similar  to 


RINGS  OF  SATURN. 


357 


that  of  the  earth  around  the  sun.  The  ring  of  Saturn  is 
inclined  about  27°  to  the  plane  of  its  orbit.  Conse- 
quently, as  the  planet  revolves  around  the  sun,  there  is  a 
change  in  the  direction  in  which  the  sun  shines  upon  it 
similar  to  that  which  produces  the  change  of  seasons  upon 
the  earth,  as  shown  in  Fig.  46,  page  109. 

The  corresponding  changes  for  Saturn  are  shown  in 
Fig.   97.     During  each  revolution  of  Saturn  the  plane 


FIG.  97. 


-DIFFERENT    ASPECTS    OF    THE    RING    OF    SATURN  AS  SEEN 
FROM   THE   EARTH. 


of  the  ring  passes  through  the  sun  twice.  This  occurred 
in  the  years  1862  and  1878,  at  two  opposite  points  of  the 
orbit,  as  shown  in  the  figure.  At  two  other  points,  mid- 
way between  these,  the  sun  shines  upon  the  plane  of  the 
ring  at  its  greatest  inclination,  about  27°.  Since  the  earth 
is  little  more  than  one  tenth  as  far  from  the  sun  as  Sat- 
urn is,  an  observer  always  sees  Saturn  nearly,  but  not 
quite,  as  if  he  were  upon  the  sun.  Hence  at  certain  times 


358  ASTRONOMY. 

the  rings  of  Saturn  are  seen  edgeways,  while  at  other 
times  they  are  at  an  inclination  of  27°,  the  aspect  depend- 
ing upon  the  position  of  the  planet  in  its  orbit.  The  fol- 
lowing are  the  times  of  some  of  the  phases  : 

1878,  February  7th.— The  edge  of  the  ring  was  turned 
toward  the  sun.  It  could  then  be  seen  only  as  a  thin 
line  of  light. 

1885. — The  planet  having  moved  forward  90°,  the  south 
side  of  the  rings  may  be  seen  at  an  inclination  of  27°. 

1891,  December. — The  planet  having  moved  90°  fur- 
ther, the  edge  of  the  ring  is  again  turned  toward  the  sun. 

1899. — The  north  side  of  the  ring  is  inclined  toward  the 
sun,  and  is  seen  at  its  greatest  inclination. 

The  rings  are  extremely  thin  in  proportion  to  their  ex- 
tent. Their  form  is  much  the  same  as  if  they  were  cut 
out  of  large  sheets  of  thin  paper.  Consequently,  when 
their  edges  are  turned  toward  the  earth,  they  appear  as  a 
thin  line  of  light,  which  can  be  seen  only  with  powerful 
telescopes.  With  such  telescopes,  the  planet  appears  as  if 
it  were  pierced  through  by  a  piece  of  very  fine  wire,  the 
ends  of  which  project  on  each  side  more  than  the  diam- 
eter of  the  planet.  It  has  frequently  been  remarked  that 
this  appearance  is  seen  on  one  side  of  the  planet,  when  no 
trace  of  the  ring  can  be  seen  on  the  other. 

There  is  sometimes  a  period  of  a  few  weeks  during 
which  the  plane  of  the  ring,  extended  outward,  passes  be  - 
tween  the  sun  and  the  earth.  That  is,  the  sun  shines  on 
one  side  of  the  ring,  while  the  other  or  dark  side  is  turned 
toward  the  earth.  In  this  case,  it  seems  to  be  established 
that  only  the  edge  of  the  ring  is  visible.  If  this  be  so, 
the  substance  of  the  rings  cannot  be  transparent  to  the 
sun's  rays,  else  it  would  be  seen  by  the  light  which  passes 
through  it. 

Possible  Changes  in  the  Rings.— In  1851  OTTO  STRUVE  pro- 
pounded a  noteworthy  theory  of  changes  going  on  in  the  rings  of 
Saturn.  From  all  the  descriptions,  figures,  and  measures  given  by 
the  older  astronomers,  it  appeared  that  two  hundred  years  ago  the 


RINGS  OF  SATURN.  359 

space  between  the  planet  and  the  inner  ring  was  at  least  equal  to 
the  combined  breadth  of  the  two  rings.  At  present  this  distance 
is  less  than  one  half  of  this  breadth.  Hence  STRUVE  concluded  that 
the  inner  ring  was  widening  on  the  inside,  so  that  its  edge  had  been 
approaching  the  planet  at  the  rate  of  about  l"-3  in  a  century.  The 
space  between  the  planet  and  the  inner  edge  of  the  bright  ring  is 
now  about  4",  so  that  if  STRUVE' s  theory  were  true,  the  inner  edge 
of  the  ring  would  actually  reach  the  planet  about  the  year  2200. 
Notwithstanding  the  amount  of  evidence  which  STRUVE  cited  in 
favor  of  his  theory,  astronomers  generally  are  incredulous  respecting 
the  reality  of  so  extraordinary  a  change.  The  measurep  necessary 
to  settle  the  question  are  so  difficult  and  the  change  is  so  slow  that 
some  time  must  elapse  before  the  theory  can  be  established,  even  if 
it  is  true.  The  measures  of  KAISER  render  this  doubtful. 

Shadow  of  Planet  and  Ring.— With  any  good  telescope  it  is 
easy  to  observe  both  the  shadow  of  the  ring  upon  the  ball  of  Saturn 
and  that  of  the  ball  upon  the  ring.  The  form  which  the  shadows 
present  often  appears  different  from  that  which  the  shadow  ought 
to  have  according  to  the  geometrical  conditions.  These  differences 
probably  arise  from  irradiation  and  other  optical  illusions. 

Constitution  of  the  Rings  of  Saturn.— The  nature  of  these 
objects  has  been  a  subject  both  of  wonder  and  of  investigation  by 
mathematicians  and  astronomers  ever  since  they  were  discovered. 
They  were  at  first  supposed  to  be  solid  bodies  ;  indeed,  from  their 
appearance  it  was  difficult  to  conceive  of  them  as  anything  else. 
The  question  then  arose  :  What  keeps  them  from  falling  on  the 
planet  ?  It  was  shown  by  LA  PLACE  that  a  homogeneous  and  solid 
ring  surrounding  the  planet  could  not  remain  in  a  state  of  equili- 
brium, but  must  be  precipitated  upon  the  central  ball  by  the  small- 
est disturbing  force.  HERSCHEL  having  thought  that  he  saw  cer- 
tain irregularities  in  the  figure  of  the  ring,  LA  PLACE  concluded  that 
the  object  could  be  kept  in  equilibrium  by  them.  He  simply  as- 
sumed this,  but  did  not  attempt  to  prove  it. 

About  1850  the  investigation  was  again  begun  by  Professors  BOND 
and  PEIRCE,  of  Cambridge.  The  former  supposed  that  the  rings 
could  not  be  solid  at  all,  because  they  had  sometimes  shown  signs  of 
being  temporarily  broken  up  into  a  large  number  of  concentric 
rings.  Although  this  was  probably  an  optical  illusion,  he  concluded 
that  the  rings  must  be  liquid.  Professor  PEIRCE  took  up  the  prob- 
lem where  LA  PLACE  had  left  it,  and  showed  that  even  an  irregular 
solid  ring  would  not  be  in  equilibrium  about  Saturn.  He  therefore 
adopted  the  view  of  BOND,  that  the  rings  were  fluid  ;  but  finding 
that  even  a  fluid  ring  would  be  unstable  without  a  support,  he  sup- 
posed that  such  a  support  might  be  furnished  by  the  satellites. 
This  view  has  also  been  abandoned. 

It  is  now  established  beyond  reasonable  doubt  that  the  rings  do 
not  form  a  continuous  mass,  but  are  really  a  countless  multitude  of 
small  separate  particles,  each  of  which  revolves  on  its  own  account. 
These  satellites  are  individually  far  too  small  to  be  seen  in  any  tele- 
scope, but  so  numerous  that  when  viewed  from  the  distance  of  the 
earth  they  appear  as  a  continuous  mass,  like  particles  of  dust  float- 


360 


ASTRONOMY. 


ing  in  a  sunbeam.  This  theory  was  first  propounded  by  CASSINI, 
of  Paris,  in  1715.  It  had  been  forgotten  for  a  century  or  more, 
when  it  was  revived  by  Profess&r  CLERK  MAXWELL  in  1856.  The 
latter  published  a  profound  mathematical  discussion  of  the  whole 
question,  in  which  he  shows  that  Ihis  hypothesis  and  this  alone 
would  account  for  the  appearances  presented  by  the  rings. 

KAISER'S  measures  of  the  dimensions  of  the  Saturuian  system  are  : 

BALL    OF     SATURN. 

Equatorial  diameter 17-"274 

Polar       •  "        15-"392 

RINGS. 

Major  axis  of  outer  ring 39*"471 

"  the  great  division 34*"227 

"       "      "  the  inner  edge  of  ring 27-"859 

Width  of  the  ring 5-806 

Dark  space  between  ball  and  ring 5 -"292 


§  3.     SATELLITES   OF   SATURN. 

Outside  the  rings  of  Saturn  revolve  its  eight  satellites, 
the  order  and  discovery  of  which  are  shown  in  the  following 
table  : 


Distance 

No. 

NAME. 

from 

Discoverer. 

Date  of  Discovery. 

Planet. 

1 

Mimas. 

3-3 

Herschel. 

1789,  September  17. 

2 

Enceladus. 

4-3 

Herschel. 

1789,  August  28. 

3 

Tethys. 

5-3 

Cassini. 

1684,  March. 

4 

Dione. 

6-8 

Cassini. 

1684,  March. 

5 

Rhea. 

9-5 

Cassini. 

1672,  December  23. 

6 

Titan. 

20-7 

Huyghens. 

1655,  March  25. 

7 

Hyperion. 

26-8 

Bond. 

1848,  September  16. 

8 

Japetus. 

64-4 

Cassini. 

1671,  October. 

The  distances  from  the  planet  are  given  in  radii  of  the 
latter.  The  satellites  Mimas  and  Hyperion  are  visible 
only  in  the  most  powerful  telescopes.  The  brightest  of 
all  is  Titan,  which  can  be  seen  in  a  telescope  of  the  small- 
est ordinary  size.  Japetus  has  the  remarkable  peculiarity 


SATELLITES  OF  SATURN. 


361 


of  appearing  nearly  as  bright  as  Titan  when  seen  west  of 
the  planet,  and  so  faint  as  to  be  visible  only  in  large  tel- 
escopes when  on  the  other  side.  This  appearance  is  ex- 
plained by  supposing  that,  like  our  moon,  it  always  pre- 
sents the  same  face  to  the  planet,  and  that  one  side  of  it  is 
black  and  the  other  side  white.  "When  west  of  the  planet, 
the  bright  side  is  turned  toward  the  earth  and  the  satellite  is 
visible.  On  the  other  side  of  the  planet,  the  dark  side  is 
turned  toward  us,  and  it  is  nearly  invisible.  Most  of  the 
remaining  five  satellites  can  be  ordinarily  seen  with  tele- 
scopes of  moderate  power. 

The  elements  of  all  the  satellites  are  shown  in  the  fol- 
lowing table  : 


SATELLITE. 

Mean  Daily 
Motion. 

Mean 
Distance 
from 
Saturn. 

Longitude 
of 
Peri-Sat. 

Eccen- 
tricity. 

Inclina- 
tion to 
Ecliptic. 

Longitude 
of 

Node 

Mimas...  . 

381-953 

9 

V 

28    00 

168    00 

Euceladus 

262-721 

V 

? 

28    00 

168    00 

Tethys.  .  . 

190-69773 

42-70 

V 

? 

28     10 

167    38 

Diane.  .  .  . 

131-534930 

54-60 

V 

? 

28     10 

167    38 

Rhea  

79-690216 

76-12 

? 

? 

28    11 

166    34 

Titan  

22-577033 

176-75 

257.16 

•0286 

27    34 

167    56 

Hyperion. 

16-914 

214-22 

40-00 

-125 

28    00 

168    00 

Japetus.  . 

4-538036 

514-64 

351-25 

•0282 

18    44 

142    53 

CHAPTER    IX. 

THE   PLANET  UKANUS. 

Uranus  was  discovered  on  March  13th,  1781,  by  Sir 
WILLIAM  HERSCHEL  (then  an  amateur  observer)  with  a 
ten-foot  reflector  made  by  himself.  He  was  examining  a 
portion  of  the  sky  near  H  Geminorum,  when  one  of  the 
stars  in  the  field  of  view  attracted  his  notice  by  its  pecu- 
liar appearance.  On  further  scrutiny,  it  proved  to  have  a 
planetary  disk,  and  a  motion  of  over  2"  per  hour.  HER- 
SCHEL at  first  supposed  it  to  be  a  comet  in  a  distant  part 
of  its  orbit,  and  under  this  impression  parabolic  orbits 
were  computed  for  it  by  various  mathematicians.  None 
of  these,  however,  satisfied  subsequent  observations, 
and  it  was  finally  announced  by  LEXELL  and  LA  PLACE 
that  the  new  body  was  a  planet  revolving  in  a  nearly 
circular  orbit.  We  can  scarcely  comprehend  now  the 
enthusiasm  with  which  this  discovery  was  received.  No 
new  body  (save  comets)  had  been  added  to  the  solar  system 
since  the  discovery  of  the  third  satellite  oi  Saturn  in  1684, 
and  all  the  major  planets  of  the  heavens  had  been  known 
for  thousands  of  years. 

HERSCHEL  suggested,  as  a  name  for  the  planet,  Geor- 
gium  Sidus,  and  even  after  1800  it  was  known  in  the  Eng- 
lish Nautical  Almanac  as  the  Georgian  Planet.  LALANDE 
suggested  Herschel  as  its  designation,  but  this  was  judged 
too  personal,  and  finally  the  name  Uranus  was  adopted. 
Its  symbol  was  for  a  time  written  Jjl,  in  recognition  of  the 
name  proposed  by  LALANDE. 

Uranus  revolves  about  the  sun  in  84  years.  Its  appar- 
ent diameter  as  seen  from  the  earth  varies  little,  being 


THE  PLANET  UHANU8.  363 

about  3". 9.  Its  true  diameter  is  about  50,000  kilometres, 
arid  its  figure  is,  so  far  as  we  yet  know,  exactly  spherical. 

In  physical  appearance  it  is  a  small  greenish  disk  with- 
out markings.  It  is  possible  that  the  centre  of  the  disk  is 
slightly  brighter  than  the  edges.  At  its  nearest  approach 
to  the  earth,  it  shines  as  a  star  of  the  sixth  magnitude, 
and  is  just  visible  to  an  acute  eye  when  the  attention  is 
directed  to  its  place.  In  small  telescopes  with  low  pow- 
ers, its  appearance  is  not  markedly  different  from  that  of 
stars  of  about  its  own  brilliancy. 

It  is  customary  to  speak  of  HERSCHEL'S  discovery  of 
Uranus  as  an  accident ;  but  this  is  not  entirely  just,  as 
all  conditions  for  the  detection  of  such  an  object,  if  it  ex- 
isted, were  fulfilled.  At  the  same  time  the  early  identifi- 
cation of  it  as  a  planet  was  more  easy  than  it  would  have 
been  eleven  days  earlier,  when,  as  ARAGO  points  out,  the 
planet  was  stationary. 

Sir  WILLIAM  HERSCHEL  suspected  that  Uranus  was  ac- 
companied by  six  satellites. 

Of  the  existence  of  two  of  these  satellites  there  has 
never  been  any  doubt,  as  they  were  steadily  observed  by 
HERSCHEL  from  1787  until  1810,  and  by  Sir  JOHN  HER- 
SCHEL during  the  years  1828  to  1832,  as  well  as  by  other 
later  observers.  None  of  the  other  four  satellites  de- 
scribed by  HERSCHEL  have  ever  been  seen  by  other  ob- 
servers, and  he  was  undoubtedly  mistaken  in  supposing 
them  to  exist.  Two  additional  ones  were  discovered  by 
LASSELL  in  1847,  and  are,  with  the  satellites  of  Mars,  the 
faintest  objects  in  the  solar  system.  Neither  of  them  is 
identical  with  any  of  the  missing  ones  of  HERSCHEL.  As 
Sir  WILLIAM  HERSCHEL  had  suspected  six  satellites,  the 
following  names  for  the  true  satellites  are  generally  adopt- 
ed to  avoid  confusion  : 

DATS. 

I,  Ariel Period  =    2-520383 

II,    Umbriel "     =    4-144181 

III,   Titania,  HERSCHEL'S  (II.) "     =    8-705897 

IY,   Oleron,  HERSCHEL'S  (IV.).  . ....'"     =  13-463269 


364  ASTRONOMY. 

It  is  an  interesting  question  whether  the  observations 
which  HERSCH EL  assigned  to>  his  supposititious  satellite  I 
may  not  be  composed  of  observations  sometimes  of  Ariel, 
sometimes  of  Umbriel.  In  fact,  out  of  nine  supposed 
observations  of  I,  one  case  alone  was  noted  by  HEKSCHEL 
in  which  his  positions  were  entirely  trustworthy,  and  on 
this  night  Umbriel  was  in  the  position  of  his  supposed 
satellite  I. 

It  is  likely  that  Ariel  varies  in  brightness  on  different 
sides  of  the  planet,  and  the  same  phenomenon  has  also 
been  suspected  for  Titania. 

The  most  remarkable  feature  of  the  satellites  of  Uranus  is  that 
their  orbits  are  nearly  perpendicular  to  the  ecliptic  instead  of 
having  a  small  inclination  to  that  plane,  like  those  of  all  the  orbits 
of  both  planets  and  satellites  previously  known.  To  form  a  correct 
idea  of  the  position  of  the  orbits,  we  must  imagine  them  tipped  over 
until  their  north  pole  is  nearly  8°  below  the  ecliptic,  instead  of  90° 
above  it.  The  pole  of  the  orbit  which  should  be  considered  as  the 
north  one  is  that  from  which,  if  an  observer  look  down  upon  a  re- 
volving body,  the  latter  would  seem  to  turn  in  a  direction  opposite 
that  of  the  hands  of  a  watch.  When  the  orbit  is  tipped  over  more 
than  a  right  angle,  the  motion  from  a  point  in  the  direction  of  the 
north  pole  of  the  ecliptic  will  seem  to  be  the  reverse  of  this  ;  it  is 
therefore  sometimes  considered  to  be  retrograde.  This  term  is  fre- 
quently applied  to  the  motion  of  the  satellites  of  Uranus,  but  is 
rather  misleading,  since  the  motion,  being  nearly  perpendicular  to 
the  ecliptic,  is  not  exactly  expressed  by  the  term. 

The  four  satellites  move  in  the  same  plane,  so  far  as  the  most  re- 
fined observations  have  ever  shown.  This  fact  renders  it  highly 
probable  that  the  planet  Uranus  revolves  on  its  axis  in  the  same 
plane  with  the  orbits  of  the  satellites,  and  is  therefore  an  oblate 
spheroid  like  the  earth.  This  conclusion  is  founded  on  the  consid- 
eration that  if  the  planes  of  the  satellites  were  not  kept  together  by 
some  cause,  they  would  gradually  deviate  from  each  other  owing  to 
the  attractive  force  of  the  sun  upon  the  planet.  The  different  satel- 
lites would  deviate  by  different  amounts,  and  it  would  be  extremely 
improbable  that  all  the  orbits  would  at  any  time  be  found  in  the 
same  plane.  Since  we  see  them  in  the  same  plane,  we  conclude  that 
some  force  keeps  them  there,  and  the  oblateness  of  the  planet  would 
cause  such  a  force. 


CHAPTER    X. 

THE   PLANET   NEPTUNE. 

AFTER  the  planet  Uranus  had  been  observed  for  some 
thirty  years,  tables  of  its  motion  were  prepared  by 
BOUVARD.  He  had  as  data  available  for  this  purpose  not 
only  the  observations  since  1781,  but  also  observations 
made  by  LE  MONNIER,  FLAMSTEED,  and  others,  extending 
back  as  far  as  1695,  in  which  the  planet  was  observed  for 
a  fixed  star  and  so  recorded  in  their  books.  As  one  of 
the  chief  difficulties  in  the  way  of  obtaining  a  theory  of 
the  planet's  motion  was  the  short  period  of  time  during 
which  it  had  been  regularly  observed,  it  was  to  be  sup- 
posed that  these  ancient  observations  would  materially  aid 
in  obtaining  exact  accordance  between  the  theory  and  ob- 
servation. But  it  was  found  that,  after  allowing  for  all 
perturbations  produced  by  the  known  planets,  the  ancient 
and  modern  observations,  though  undoubtedly  referring  to 
the  same  object,  were  yet  not  to  be  reconciled  with  each 
other,  but  differed  systematically.  BOUVARD  was  forced 
to  omit  the  older  observations  in  his  tables,  which  were 
published  in  1820,  and  to  found  his  theory  upon  the 
modern  observations  alone.  By  so  doing,  he  obtained  a 
good  agreement  between  theory  and  the  observations  of 
the  few  years  immediately  succeeding  1820. 

BOUVARD  seems  to  have  formulated  the  idea  that  a  possi- 
ble cause  for  the  discrepancies  noted  might  be  the  exist- 
ence of  an  unknown  planet,  but  the  meagre  data  at  his 
disposal  forced  him  to  leave  the  subject  untouched.  In 
1830  it  was  found  that  the  tables  which  represented  the 


366  ASTRONOMY. 

motion  of  the  planet  well  in  1820-25  were  20"  in  error,  in 
1840  the  error  was  90",  and  in  1845  it  was  over  120". 

These  progressive  and  systematic  changes  attracted  the 
attention  of  astronomers  to  the  subject  of  the  theory  of 
the  motion  of  Uranus.  The  actual  discrepancy  (120")  in 
1845  was  not  a  quantity  large  in  itself.  Two  stars  of  the 
magnitude  of  Uranus,  and  separated  by  only  120",  would 
be  seen  as  one  to  the  unaided  eye.  It  was  on  account  of 
its  systematic  and  progressive  increase  that  suspicion  was 
excited.  Several  astronomers  attacked  the  problem  in  vari- 
ous ways.  The  elder  STRUVE,  at  Pulkova,  prosecuted  a 
search  for  a  new  planet  along  with  his  double  star  obser- 
vations ;  BESSEL,  at  Koenigsberg,  set  a  student  of  his  own, 
FLEMING,  at  a  new  comparison  of  observation  with  theo- 
ry, in  order  to  furnish  data  for  a  new  determination  ; 
ARAGO,  then  Director  of  the  Observatory  at  Paris,  sug- 
gested this  subject  in  1845  as  an  interesting  field  of  re- 
search to  LE  YERRIER,  then  a  rising  mathematician 
and  astronomer.  Mr.  J.  C.  ADAMS,  a  student  in  Cam- 
bridge University,  England,  had  become  aware  of  the 
problems  presented  by  the  anomalies  in  the  motion  of 
Uranus,  and  had  attacked  this  question  as  early  as  1843. 
In  October,  1845,  ADAMS  communicated  to  the  Astrono- 
mer Koyal  of  England  elements  of  a  new  planet  so  situated 
as  to  produce  the  perturbations  of  the  motion  of  Uranus 
which  had  actually  been  observed.  Such  a  prediction 
from  an  entirely  unknown  student,  as  ADAMS  then  was, 
did  not  carry  entire  conviction  with  it.  A  series  of  acci- 
dents prevented  the  unknown  planet  being  looked  for  by 
one  of  the  largest  telescopes  in  England,  and  so  the  mat- 
ter apparently  dropped.  It  may  be  noted,  however,  that 
we  now  know  ADAMS'  elements  of  the  new  planet  to  have 
been  so  near  the  truth  that  if  it  had  been  really  looked  for 
by  the  powerful  telescope  which  afterward  discovered  its 
satellite,  it  could  scarcely  have  failed  of  detection. 

BESSEL'S  pupil  FLEMING  died  before  his  work  was  done, 
and  BESSEL'S  researches  were  temporarily  brought  to 


DISCOVERT  OF  NEPTUNE.  367 

an  end.  STRUVE'S  search  was  unsuccessful.  Only  LE 
VERRIER  continued  his  investigations,  and  in  the  most 
thorough  manner.  He  first  computed  anew  the  pertur- 
bations of  Uranus  produced  by  the  action  of  Jupiter  and 
Saturn.  Then  he  examined  the  nature  of  the  irregulari- 
ties observed.  These  showed  that  if  they  were  caused  by 
an  unknown  planet,  it  could  not  be  between  Saturn  and 
Uranus,  or  else  Saturn  would  have  been  more  affected 
than  was  the  case. 

The  new  planet  was  outside  of  Uranus  if  it  existed  at 
all,  and  as  a  rough  guide  BODE'S  law  was  invoked,  which 
indicated  a  distance  about  twice  that  of  Uranus.  In  the 
summer  of  1846,  LE  YEKRIEB  obtained  complete  elements 
of  a  new  planet,  which  would  account  for  the  observed 
irregularities  in  the  motion  of  Uranus,  and  these  were 
published  in  France.  They  were  very  similar  to  those  of 
ADAMS,  which  had  been  communicated  to  Professor  CHAL- 
LIS,  the  Director  of  the  Observatory  of  Cambridge. 

A  search  was  immediately  begun  by  CHALLIS  for  such 
an  object,  and  as  no  star-maps  were  at  hand  for  this  region 
of  the  sky,  he  began  mapping  the  surrounding  stars.  In 
so  doing  the  new  planet  was  actually  observed,  both  on 
August  4th  and  12th,  1846,  but  the  observations  remain- 
ing unreduced,  and  so  the  planetary  nature  of  the  object 
was  not  recognized. 

In  September  of  the  same  year,  LE  TERRIER  wrote  to 
Dr.  GALLE,  then  Assistant  at  the  Observatory  of  Berlin, 
asking  him  to  search  for  the  new  planet,  and  directing 
him  to  the  place  where  it  should  be  found.  By  the  aid 
of  an  excellent  star  chart  of  this  region,  which  had  just 
been  completed  by  Dr.  BREMIKER,  the  planet  was  found 
September  23d,  1846. ' 

The  strict  rights  of  discovery  lay  with  LE  VERRIER, 
but  the  common  consent  of  mankind  has  always  credited 
ADAMS  with  an  equal  share  in  the  honor  attached  to  this 
most  brilliant  achievement.  Indeed,  it  was  only  by  the 
most  unfortunate  succession  of  accidents  that  the  discovery 


368  ASTRONOMY. 

did  not  attach  to  ADAMS'  researches.  One  thing  must  in 
fairness  be  said,  and  that  is  .that  the  results  of  LE  YER- 
RIEB,  which  were  reached  after  a  most  thorough  investi- 
gation of  the  whole  ground,  were  announced  with  an  en- 
tire confidence,  which,  perhaps,  was  lacking  in  the  other 
case. 

This  brilliant  discovery  created  more  enthusiasm  than 
even  the  discovery  of  Uranus,  as  it  was  by  an  exercise  of 
far  higher  qualities  that  it  was  achieved.  It  appeared  to 
savor  of  the  marvellous  that  a  mathematician  could  say 


FIG.  98. 

to  a  working  astronomer  that  by  pointing  his  telescope  to 
a  certain  small  area,  within  it  should  be  found  a  new 
major  planet.  Yet  so  it  was. 

The  general  nature  of  the  disturbing  force  which  re- 
vealed the  new  planet  may  be  seen  by  Fig.  98,  which 
shows  the  orbits  of  the  two  planets,  and  their  respective 
motions  between  1781  and  1840.  The  inner  orbit  is  that 
of  Uranus,  the  outer  one  that  of  Neptune.  The  arrows 
passing  from  the  former  to  the  latter  show  the  directions 
of  the  attractive  force  of  Neptune.  It  will  be  seen  that 


SATELLITE  OF  NEPTUNE.  369 

the  two  planets  were  in  conjunction  in  the  year  1822. 
Since  that  time  Urcmus  has,  by  its  more  rapid  motion, 
passed  more  than  90°  beyond  Neptune,  and  will  continue 
to  increase  its  distance  from  the  latter  until  the  begin- 
ning of  the  next  century. 

Our  knowledge  regarding  Neptune  is  mostly  confined 
to  a  few  numbers  representing  the  elements  of  its  motion. 
Its  mean  distance  is  more  than  4,000,000,000  kilometres 
(2,775,000,000  miles) ;  its  periodic  time  is  164-78  years  ; 
its  apparent  diameter  is  2" -6  seconds,  corresponding  to  a 
true  diameter  of  55,000  kilometres.  Gravity  at  its  surface 
is  about  nine  tenths  of  the  corresponding  terrestrial  surface 
gravity.  Of  its  rotation  and  physical  condition  nothing 
is  known.  Its  color  is  a  pale  greenish  blue.  It  is  attend- 
ed by  one  satellite,  the  elements  of  whose  orbit  are  given 
herewith.  It  was  discovered  by  Mr.  LASSELL,  of  Eng- 
land, in  1847.  It  is  about  as  faint  as  the  two  outer  satel- 
lites of  Uranus,  and  requires  a  telescope  of  twelve  inches 
aperture  or  upward  to  be  well  seen. 

ELEMENTS   OF  THE   SATELLITE  OP  NEPTUNE,  FROM  WASHINGTON 
OBSERVATIONS. 

Mean  Daily  Motion 61° -25679 

Periodic  Time 5d-87690 

Distance  (log.  A  =  1-47814) 16"-275 

Inclination  of  Orbit  to  Ecliptic 145°      V 

Longitude  of  Node  (1850) 184°    30' 

Increase  in  100  Years. .  1°    24' 


The  great  inclination  of  the  orbit  shows  that  it  is  turned  nearly 
upside  down  ;  the  direction  of  motion  is  therefore  retrogade. 


CHAPTER  XI. 

THE   PHYSICAL   CONSTITUTION  OF   THE 
PLANETS. 

IT  is  remarkable  that  the  eight  large  planets  of  the  solar 
system,  considered  with  respect  to  their  physical  constitu- 
tion as  revealed  by  the  telescope  and  the  spectroscope, 
may  be  divided  into  four  pairs,  the  planets  of  each  pair 
having  a  great  similarity,  and  being  quite  different  from 
the  adjoining  pair.  Among  the  most  complete  and  sys- 
tematic studies  of  the  spectra  of  all  the  planets  are  those 
made  by  Mr.  HUGGINS,  of  London,  and  Dr.  VOGEL,  of 
Berlin.  In  what  we  have  to  say  of  the  results  of  spectro- 
scopy,  we  shall  depend  entirely  upon  the  reports  of  these 
observers. 

Mercury  and  Venus. — Passing  outward  from  the  sun, 
the  first  pair  we  encounter  will  be  Mercury  and  Venus. 
The  most  remarkable  feature  of  these  two  planets  is  a  neg- 
ative rather  than  a  positive  one,  being  the  entire  absence 
of  any  certain  evidence  of  change  on  their  surfaces.  We 
have  already  shown  that  Venus  has  a  considerable  atmos- 
phere, while  there  is  no  evidence  of  any  such  atmosphere 
around  Mercury.  They  have  therefore  not  been  proved 
alike  in  this  respect,  yet,  on  the  other  hand,  they  have  not 
been  proved  different.  In  every  other  respect  than  this, 
the  similarity  appears  perfect.  No  permanent  markings 
have  ever  been  certainly  seen  on  the  disk  of  either.  If, 
as  is  possible,  the  atmosphere  of  both  planets  is  filled  with 
clouds  and  vapor,  110  change,  no  openings,  and  no  for- 


PHYSICAL  CONSTITUTION  OF  THE  PLANETS.     371 

mations  among  these  cloud  masses  are  visible  from  the 
earth.  Whenever  either  of  these  planets  is  in  a  certain 
position  relative  to  the  earth  and  the  sun,  it  seemingly 
presents  the  same  appearance,  and  not  the  slightest 
change  occurs  in  that  appearance  from  the  rotation  of  the 
planet  on  its  axis,  which  every  analogy  of  the  solar  sys- 
tem leads  us  to  believe  must  take  place. 

When  studied  with  the  spectroscope,  the  spectra  of 
Mercury  and  Venus  do  not  differ  strikingly  from  that  of 
the  sun.  This  would  seem  to  indicate  that  the  atmos- 
pheres of  these  planets  do  not  exert  any  decided  absorption 
upon  the  rays  of  light  which  pass  through  them  ;  or,  at 
least,  they  absorb  only  the  same  rays  which  are  absorbed 
by  the  atmosphere  of  the  sun  and  by  that  of  the  earth. 
The  one  point  of  difference  which  Dr.  VOGEL  brings  out 
is,  that  the  lines  of  the  spectrum  produced  by  the  absorp- 
tion of  our  own  atmosphere  appear  darker  in  the  spectrum 
of  Venus.  If  this  were  so,  it  would  indicate  that  the  at- 
mosphere of  Venus  is  similar  in  constitution  to  that  of 
our  earth,  because  it  absorbs  the  same  rays.  But  the 
means  of  measuring  the  darkness  of  the  lines  are  as  yet 
so  imperfect  that  it  is  impossible  to  speak  with  certainty 
on  a  point  like  this.  Dr.  VOGEL  thinks  that  the  light 
from  Venus  is  for  the  most  part  reflected  from  clouds  in 
the  higher  region  of  the  planet's  atmosphere,  and  there- 
fore reaches  us  without  passing  through  a  great  depth  of 
that  atmosphere. 

The  Earth  and  Mars. — These  planets  are  distinguished 
from  all  the  others  in  that  their  visible  surfaces  are  marked 
by  permanent  features,  which  show  them  to  be  solid,  and 
which  can  be  seen  from  the  other  heavenly  bodies.  It  is 
true  that  we  cannot  study  the  earth  from  any  other  body, 
but  we  can  form  a  very  correct  idea  how  it  would  look  if 
seen  in  this  way  (from  the  moon,  for  instance).  Wherever 
the  atmosphere  was  clear,  the  outlines  of  the  continents 
and  oceans  would  be  visible,  while  they  would  be  invisible 
where  the  air  was  cloudy. 


372  ASTRONOMY. 

Now,  so  far  as  we  can  judge  from  observations  made 
at  so  great  a  distance,  never  much  less  than  forty  mil- 
lions of  miles,  the  planet  Mars  presents  to  our  tele- 
scopes very  much  the  same  general  appearance  that  the 
earth  would  if  observed  from  an  equally  great  distance. 
The  only  exception  is  that  the  visible  surface  of  Mars  is 
seemingly  much  less  obscured  by  clouds  than  that  of  the 
earth  would  be.  In  other  words,  that  planet  has  a  more 
sunny  sky  than  ours.  It  is,  of  course,  impossible  to  say 
what  conditions  we  might  find  could  we  take  a  much 
closer  view  of  Mars :  all  we  can  assert  is,  that  so  far  as 
we  can  judge  from  this  distance,  its  surface  is  like  that  of 
the  earth. 

This  supposed  similarity  is  strengthened  by  the  spectro- 
scopic  observations.  The  lines  of  the  spectrum  due  to 
aqueous  vapor  in  our  atmosphere  are  found  by  Dr.  YOGEL 
to  be  so  much  stronger  in  Mars  as  to  indicate  an  absorp- 
tion by  such  vapor  in  its  atmosphere.  Dr.  HUGGINS  had 
previously  made  a  more  decisive  observation,  having 
found  a  well-marked  line  to  which  there  is  no  correspond- 
ing strong  line  in  the  solar  spectrum.  This  would  indi- 
cate that  the  atmosphere  of  Mars  contains  some  element 
not  found  in  our  own,  but  the  observations  are  too  diffi- 
cult to  allow  of  any  well-established  theory  being  yet 
built  upon  them. 

Jupiter  and  Saturn. — The  next  pair  of  planets  are 
Jupiter  and  Saturn.  Their  peculiarity  is  that  no  solid 
crust  or  surface  is  visible  from  without.  In  this  respect 
they  differ  from  the  earth  and  Mars,  and  resemble  Mer- 
cury and  Venus.  But  they  differ  from  the  latter  in  the 
very  important  point  that  constant  changes  can  be  seen 
going  on  at  their  surfaces.  The  nature  of  these  changes 
has  been  discussed  so  fully  in  treating  of  these  planets  in- 
dividually, that  we  need  not  go  into  it  more  fully  at  pres- 
ent. It  is  sufficient  to  say  that  the  preponderance  of  evi- 
dence is  in  favor  of  the  view  that  these  planets  have  no 
solid  crusts  whatever,  but  consist  of  masses  of  molten 


PHYSICAL  CONSTITUTION  OF  THE  PLANETS.     373 

matter,  surrounded  by  envelopes  of  vapor  constantly  rising 
from  the  interior. 

The  view  that  the  greater  part  of  the  apparent  volume  of 
these  planets  is  made  of  a  seething  mass  of  vapor  is  further 
strengthened  by  their  very  small  specific  gravity.  This 
can  be  accounted  for  by  supposing  that  the  liquid  interior 
is  nothing  more  than  a  comparatively  small  central  core, 
and  that  the  greater  part  of  the  bulk  of  each  planet  is 
composed  of  vapor  of  small  density. 

That  the  visible  surfaces  of  Jupiter  and  Saturn  are  cov- 
ered by  some  kind  of  an  atmosphere  follows  not  only  from 
the  motion  of  the  cloud  forms  seen  there,  but  from  the 
spectroscopic  observations  of  HUGGINS  in  1864.  He 
found  visible  absorption-bands  near  the  red  end  of  the 
spectrum  of  each  of  these  planets.  YOGEL  found  a  com- 
plete similarity  between  the  spectra  of  the  two  planets, 
the  most  marked  feature  being  a  dark  band  in  the  red. 
What  is  worthy  of  remark,  though  not  at  all  surprising,  is 
that  this  band  is  not  found  in  the  spectrum  of  Saturn's 
rings.  This  is  what  we  should  expect,  as  it  is  hardly  pos- 
sible that  these  rings  should  have  any  atmosphere,  owing 
to  their  very  small  mass.  An  atmosphere  on  bodies  of  so 
slight  an  attractive  power  would  expand  away  by  its  own 
elasticity  and  be  all  attracted  around  the  planet. 

Uranus  and  Neptune. — These  planets  have  a  strikingly 
similar  aspect  when  seen  through  a  telescope.  They 
differ  from  Jupiter  and  Saturn  in  that  no  changes  or  va- 
riations of  color  or  aspect  can  be  made  out  upon  their  sur- 
faces ;  and  from  the  earth  and  Mars  in  the  absence  of  any 
permanent  features.  Telescopically,  therefore,  we  might 
classify  them  with  Mercury  and  Venus,  but  the  spectro- 
scope reveals  a  constitution  entirely  different  from  that  of 
any  other  planets.  The  most  marked  features  of  their 
spectra  are  very  dark  bands,  evidently  produced  by  the 
absorption  of  dense  atmospheres.  Owing  to  the  extreme 
faintness  of  the  light  which  reaches  us  from  these  distant 
bodies,  the  regular  lines  of  the  solar  spectrum  are  entirely 


374 


ASTRONOMY. 


invisible  in  their  spectra,  yet  these  dark  bands  which  are 
peculiar  to  them  have  been  seen  by  HUGGINS,  SECCHI, 
pl^^mmMH^MH  VOGEL,  and  perhaps  others. 

This  classification  of  the 
eight  planets  into  pairs  is  ren- 
dered yet  more  striking  by 
the  fact  that  it  applies  to 
what  we  have  been  able  to 
discover  respecting  the  rota- 
tions of  these  bodies.  The 
rotation  of  the  inner  pair, 
Mercury  and  Venus,  has 
eluded  detection,  notwith- 
standing their  comparative 
proximity  to  us.  The  next 
pair,  the  earth  and  Mars, 
have  perfectly  definite  times 
of  rotation,  because  their 
outer  surfaces  consist  of  solid 
crusts,  every  part  of  which 
must  rotate  in  the  same  time. 
The  next  pair,  Jupiter  and 
Saturn,  have  well-established 
times  of  rotation,  but  these 
times  are  not  perfectly  defi- 
nite, because  the  surfaces  of 
these  planets  are  not  solid, 
and  different  portions  of  their 
mass  may  rotate  in  slightly 
different  times.  Jupiter  and 
Saturn  have  also  in  common 


FlG.  99. — SPECTRUM  OP  URANUS. 


a  very  rapid  rate  of  rotation.  Finally,  the  outer  pair,  Ura- 
nus and  Neptune,  seem  to  be  surrounded  by  atmospheres  of 
such  density  that  no  evidence  of  rotation  can  be  gathered. 
Thus  it  seems  that  of  the  eight  planets,  only  the  central 
four  have  yet  certainly  indicated  a  rotation  on  their  axes. 


CHAPTER    XII. 

METEORS. 
§   1.    PHENOMENA  AND  CAUSES  OP  METEORS. 

DURING  the  present  century,  evidence  has  been  collected 
that  countless  masses  of  matter,  far  too  small  to  be  seen 
with  the  most  powerful  telescopes,  are  moving  through 
the  planetary  spaces.  This  evidence  is  afforded  by  the 
phenomena  of  ' '  aerolites, "  "  meteors, ' '  and  ( i  shooting 
stars. ' '  Although  these  several  phenomena  have  been  ob- 
served and  noted  from  time  to  time  since  the  earliest  his- 
toric era,  it  is  only  recently  that  a  complete  explanation 
has  been  reached. 

Aerolites. — Reports  of  the  falling  of  large  masses  of 
stone  or  iron  to  the  earth  have  been  familiar  to  antiqua- 
rian students  for  many  centuries.  AKAGO  has  collected 
several  hundred  of  these  reports.  In  one  instance  a  monk 
wras  killed  by  the  fall  of  one  of  these  bodies.  One  or  two 
other  cases  of  death  from  this  cause  are  supposed  to  have 
occurred.  Notwithstanding  the  number  of  instances  on 
record,  aerolites  fall  at  such  wide  intervals  as  to  be  ob- 
served by  very  few  people,  consequently  doubt  was  fre- 
quently cast  upon  the  correctness  of  the  narratives.  <  The 
problem  where  such  a  body  could  come  from,  or  how  it 
could  get  into  the  atmosphere  to  fall  down  again,  formerly 
seemed  so  nearly  incapable  of  solution  that  it  required 
some  credulity  to  admit  the  facts.  When  the  evidence 
became  so  strong  as  to  be  indisputable,  theories  of  their 
origin  began  to  be  propounded.  One  theory  quite  fashion- 


376  ASTRONOMY. 

able  in  the  early  part  of  this  century  was  that  they  were 
thrown  from  volcanoes  jn  the  moon.  This  theory, 
though  the  subject  of  mathematical  investigation  by  LA 
PLACE  and  others,  is  now  no  longer  thought  of. 

The  proof  that  aerolites  did  really  fall  to  the  ground 
first  became  conclusive  by  the  fall  being  connected  with 
other  more  familiar  phenomena.  Nearly  every  one  who 
is  at  all  observant  of  the  heavens  is  familiar  with  bolides^ 
or  fire-balls — brilliant  objects  having  the  appearance  of 
rockets,  which  are  occasionally  seen  moving  with  great  ve- 
locity through  the  upper  regions  of  the  atmosphere. 
Scarcely  a  year  passes  in  which  such  a  body  of  extraordi- 
nary brilliancy  is  not  seen.  Generally  these  bodies,  bright 
though  they  may  be,  vanish  without  leaving  any  trace,  or 
making  themselves  evident  to  any  sense  but  that  of  sight. 
But  on  rare  occasions  their  appearance  is  followed  at  an 
interval  of  several  minutes  by  loud  explosions  like  the  dis- 
charge of  a  battery  of  artillery.  On  still  rarer  occasions, 
masses  of  matter  fall  to  the  ground.  It  is  now  fully 
understood  that  the  fall  of  these  aerolites  is  always  ac- 
companied by  light  and  sound,  though  the  light  may  be 
invisible  in  the  daytime. 

When  chemical  analysis  was  applied  to  aerolites,  they 
were  proved  to  be  of  extramundane  origin,  because  they 
contained  chemical  combinations  not  found  in  terrestrial 
substances.  It  is  true  that  they  contained  no  new  chemi- 
cal elements,  but  only  combination  of  the  elements  which 
are  found  on  the  earth.  These  combinations  are  now  so 
familiar  to  mineralogists  that  they  can  distinguish  an 
aerolite  from  a  mineral  of  terrestrial  origin  by  a  careful 
examination.  One  of  the  largest  components  of  these 
bodies  is  iron.  Specimens  having  very  much  the  appear- 
ance of  great  masses  of  iron  are  found  in  the  National 
Museum  at  Washington. 

Meteors. — Although  the  meteors  we  have  described  are 
of  dazzling  brilliancy,  yet  they  run  by  insensible  grada- 
tions into  phenomena,  which  any  one  can  see  on  any  clear 


CAUSE  OF  METEORS.  377 

night.  The  most  brilliant  meteors  of  all  are  likely  to  be 
seen  by  one  person  only  two  or  three  times  in  his  life. 
Meteors  having  the  appearance  and  brightness  of  a  distant 
rocket  may  be  seen  several  times  a  year  by  any  one  in  the 
habit  of  walking  out  during  the  evening  and  watching  the 
sky.  Smaller  ones  occur  more  frequently  ;  and  if  a  care- 
ful watch  be  kept,  it  will  be  found  that  several  of  the 
faintest  class  of  all,  familiarly  known  as  shooting  stars,  can 
be  seen  on  every  clear  night.  We  can  draw  no  distinction 
between  the  most  brilliant  meteor  illuminating  the  whole 
sky,  and  perhaps  making  a  noise  like  thunder,  and  the 
faintest  shooting  star,  except  one  of  degree.  There  seems 
to  be  every  gradation  between  these  extremes,  so  that  all 
should  be  traced  to  some  common  cause. 

Cause  of  Meteors. — There  is  now  no  doubt  that  all  thees 
phenomena  have  a  common  origin,  being  due  to  the  earth 
encountering  innumerable  small  bodies  in  its  annual  course 
around  the  sun.  The  great  difficulty  in  connecting  mete- 
ors with  these  invisible  bodies  arises  from  the  brilliancy 
and  rapid  disappearance  of  the  meteors.  The  question 
may  be  asked  why  do  they  burn  with  so  great  an  evolu- 
tion of  light  on  reaching  our  atmosphere  ?  To  answer  this 
question,  we  must  have  recourse  to  the  mechanical  theory 
of  heat.  It  is  now  known  that  heat  is  really  a  vibratory 
motion  in  the  particles  of  solid  bodies  and  a  progressive 
motion  in  those  of  gases.  By  making  this  motion  more 
rapid,  we  make  the  body  warmer.  By  simply  blowing  air 
against  any  combustible  body  with  sufficient  velocity,  it 
can  be  set  on  fire,  and,  if  incombustible,  the  body  will  be 
made  red-hot  and  finally  melted.  Experiments  to  deter- 
mine the  degree  of  temperature  thus  produced  have  been 
made  by  Sir  WILLIAM  THOMPSON,  who  finds  that  a  veloci- 
ty of  about  50  metres  per  second  corresponds  to  a  rise  of 
temperature  of  one  degree  Centigrade.  From  this  the 
temperature  due  to  any  velocity  can  be  readily  calculated 
on  the  principle  that  the  increase  of  temperature  is  pro- 
portional to  the  "energy"  of  the  particles,  which  again 


378  ASTRONOMY. 

is  proportional  to  the  square  of  the  velocity.  Hence  a 
velocity  of  500  metres  per  second  would  correspond  to  a 
rise  of  100°  above  the  actual  temperature  of  the  air,  so 
that  if  the  latter  was  at  the  freezing-point  the  body  would 
be  raised  to  the  temperature  of  boiling  water.  A  velocity 
of  1500  metres  per  second  would  produce  a  red  heat.  This 
velocity  is,  however,  much  higher  than  any  that  we  can 
produce  artificially. 

The  earth  moves  around  the  sun  with  a  velocity  of 
about  30,000  metres  per  second  ;  consequently  if  it  met  a 
body  at  rest  the  concussion  between  the  latter  and  the  at- 
mosphere would  correspond  to  a  temperature  of  more  than 
300,000°.  This  would  instantly  dissolve  any  known  sub- 
stance. 

As  the  theory  of  this  dissipation  of  a  body  by  moving 
with  planetary  velocity  through  the  upper  regions  of  our 
air  is  frequently  misunderstood,  it  is  necessary  to  explain 
two  or  three  points  in  connection  with  it. 

(1.)  It  must  be  remembered  that  when  we  speak  of 
these  enormous  temperatures,  we  are  to  consider  them  as 
potential,  not  actual,  temperatures.  We  do  not  mean 
that  the  body  is  actually  raised  to  a  temperature  of  300,- 
000°,  but  only  that  the  air  acts  upon  it  as  if  it  were  put 
into  a  furnace  heated  to  this  temperature — that  is,  it  is 
rapidly  destroyed  by  the  intensity  of  the  heat. 

(2.)  This  potential  temperature  is  independent  of  the 
density  of  the  medium,  being  the  same  in  the  rarest  as  in 
the  densest  atmosphere.  But  the  actual  effect  on  the 
body  is  not  so  great  in  a  rare  as  in  a  dense  atmosphere. 
Every  one  knows  that  he  can  hold  his  hand  for  some  time 
in  air  at  the  temperature  of  boiling  water.  The  rarer  the 
air  the  higher  the  temperature  the  hand  would  bear  without 
injury.  In  an  atmosphere  as  rare  as  ours  at  the  height  of 
50  miles,  it  is  probable  that  the  hand  could  be  held  for  an 
indefinite  period,  though  its  temperature  should  be  that 
of  red-hot  iron  ;  hence  the  meteor  is  not  consumed  so  rap- 
idly as  if  it  struck  a  dense  atmosphere  with  planetary 


CA  USE  OF  METEORS.  379 

velocity.  In  the  latter  case  it  would  probably  disappear 
like  a  flash  of  lightning. 

(3.)  The  amount  of  heat  evolved  is  measured  not  by  that 
which  would  result  from  the  combustion  of  the  body,  but 
by  the  vis  viva  (energy  of  motion)  which  the  body  loses  in 
the  atmosphere.  The  student  of  physics  knows  that  mo- 
tion, when  lost,  is  changed  into  a  definite  amount  of 
heat.  If  we  calculate  the  amount  of  heat  which  is  equiv- 
alent to  the  energy  of  motion  of  a  pebble  having  a  veloc- 
ity of  20  miles  a  second,  we  shall  find  it  sufficient  to  raise 
about  1300  times  the  pebble's  weight  of  water  from  the 
freezing  to  the  boiling  point.  This  is  many  times  as  much 
heat  as  could  result  from  burning  even  the  most  combusti- 
ble body. 

(4.)  The  detonation  which  sometimes  accompanies  the 
passage  of  very  brilliant  meteors  is  not  caused  by  an  ex- 
plosion of  the  meteor,  but  by  the  concussion  produced  by 
its  rapid  motion  through  the  atmosphere.  This  concus- 
sion is  of  much  the  same  nature  as  that  produced  by  a 
flash  of  lightning.  The  air  is  suddenly  condensed  in  front 
of  the  meteor,  while  a  vacuum  is  left  behind  it. 

The  invisible  bodies  which  produce  meteors  in  the  way 
just  described  have  been  called  meteoroids.  Meteoric 
phenomena  depend  very  largely  upon  the  nature  of  the 
meteoroids,  and  the  direction  and  velocity  with  which 
they  are  moving  relatively  to  the  earth.  With  very  rare 
exceptions,  they  are  so  small  and  fusible  as  to  be  entirely 
dissipated  in  the  upper  regions  of  the  atmosphere.  Even 
of  those  so  hard  and  solid  as  to  produce  a  brilliant  light 
and  the  loudest  detonation,  only  a  small  proportion  reach 
the  earth.  It  has  sometimes  happened  that  the  meteoroid 
only  grazes  the  atmosphere,  passing  horizontally  through 
its  higher  strata  for  a  great  distance  and  continuing  its 
course  after  leaving  it.  On  rare  occasions  the  body  is  so 
hard  and  massive  as  to  reach  the  earth  without  being  en- 
tirely consumed.  The  potential  heat  produced  by  its 
passage  through  the  atmosphere  is  then  all  expended  in 


380  ASTRONOMY. 

melting  and  destroying  its  outer  layers,  the  inner  nucleus 
remaining  unchanged.  When  such  a  body  first  strikes 
the  denser  portion  of  the  atmosphere,  the  resistance  be- 
comes so  great  that  the  body  is  generally  broken  to  pieces. 
Hence  we  very  often  find  not  simply  a  single  aerolite, 
but  a  small  shower  of  them. 

Heights  of  Meteors. — Many  observations  have  been 
made  to  determine  the  height  at  which  meteors  are  seen. 
This  is  effected  by  two  observers  stationing  themselves 
several  miles  apart  and  mapping  out  the  courses  of  such 
meteors  as  they  can  observe.  In  order  to  be  sure  that  the 
same  meteor  is  seen  from  both  stations,  the  time  of  each 
observation  must  be  noted.  In  the  case  of  very  brilliant 
meteors,  the  path  is  often  determined  with  considerable 
precision  by  the  direction  in  which  it  is  seen  by  accidental 
observers  in  various  regions  of  the  country  over  which  it 
passes. 

The  general  result  from  numerous  observations  and  in- 
vestigations of  this  kind  is  that  the  meteors  and  shooting 
stars  commonly  commence  to  be  visible  at  a  height  of 
about  160  kilometres,  or  100  statute  miles.  The  separate 
results  of  course  vary  widely,  but  this  is  a  rough  mean  of 
them.  They  are  generally  dissipated  at  about  half  this 
height,  and  therefore  above  the  highest  atmosphere  which 
reflects  the  rays  of  the  sun.  From  this  it  may  be  inferred 
that  the  earth's  atmosphere  rises  to  a  height  of  at  least 
1 60  kilometres.  This  is  a  much  greater  height  than  it  was 
formerly  supposed  to  have. 

§   2.    METEORIC  SHOWERS. 

As  already  stated,  the  phenomena  of  shooting  stars  may 
be  seen  by  a  careful  observer  on  almost  any  clear  night. 
In  general,  not  more  than  three  or  four  of  them  will  be 
seen  in  an  hour,  and  these  will  be  so  minute  as  hardly  to 
attract  notice.  But  they  sometimes  fall  in  such  numbers 
as  to  present  the  appearance  of  a  meteoric  shower.  On 


METEORIC  SHOWERS.  381 

rare  occasions  the  shower  lias  been  so  striking  as  to  fill  the 
beholders  with  terror.  The  ancient  and  mediaeval  records 
contain  many  accounts  of  these  phenomena  which  have 
been  brought  to  light  through  the  researches  of  antiqua- 
rians. The  following  is  quoted  by  Professor  NEWTON 
from  an  Arabic  record  : 

"  In  the  year  599,  on  the  last  day  of  Moharrem,  stars  shot  hither 
and  thither,  and  flew  against  each  other  like  a  swarm  of  locusts  ; 
this  phenomena  lasted  until  daybreak  ;  people  were  thrown  into 
consternation,  and  made  supplication  to  the  Most  High  :  there  was 
never  the  like  seen  except  on  the  coming  of  the  messenger  of  God, 
on  whom  be  benediction  and  peace." 

It  has  long  been  known  that  some  showers  of  this  class 
occur  at  an  interval  of  about  a  third  of  a  century.  One 
was  observed  by  HUMBOLDT,  on  the  Andes,  on  the  night 
of  November  12th,  1799,  lasting  from  two  o'clock  until 
daylight.  A  great  shower  was  seen  in  this  country  in 
1833,  and  is  well  known  to  have  struck  the  negroes  of  the 
Southern  States  with  terror.  The  theory  that  the  show- 
ers occur  at  intervals  of  34  years  was  now  propounded  by 
OLBERS,  who  predicted  a  return  of  the  shower  in  1867. 
This  prediction  was  completely  fulfilled,  but  instead  of  ap- 
pearing in  the  year  1867  only,  it  was  first  noticed  in  1866. 
On  the  night  of  November  13th  of  that  year  a  remarkable 
shower  was  seen  in  Europe,  while  on  the  corresponding 
night  of  the  year  following  it  was  again  seen  in  this  coun- 
try, and,  in  fact,  was  repeated  for  two  or  three  years,  grad- 
ually dying  away. 

The  occurrence  of  a  shower  of  meteors  evidently  shows 
that  the"  earth  encounters  a  swarm  of  meteoroids.  The 
recurrence  at  the  same  time  of  the  year,  when  the  earth 
is  in  the  same  point  of  its  orbit,  shows  that  the  earth 
meets  the  swarm  at  the  same  point  in  successive  years. 
All  the  meteoroids  of  the  swarm  must  of  course  be  moving 
in  the  same  direction,  else  they  would  soon  be  widely  scat- 
tered. This  motion  is  connected  with  the  radiant  point, 
a  well-marked  feature  of  a  meteoric  shower. 


382  ASTRONOMY. 

Radiant  Point. — Suppose  that,  during  a  meteoric  shower,  we 
mark  the  path  of  each  meteor  on  a  star  map,  as  in  the  figure.  If  we 
continue  the  paths  backward  in*  a  straight  line,  we  shall  find  that 
they  all  meet  near  one  and  the  same  point  of  the  celestial  sphere  — 
that  is,  they  move  as  if  they  all  radiated  from  this  point.  The 


FlG.    100.— RADIANT  POINT  OF  METEORIC   SHOWER. 

latter  is,  therefore,  called  the  radiant  point.  In  the  figure  the  lines 
do  not  all  pass  accurately  through  the  same  point.  This  is  owing 
to  the  unavoidable  errors  made  in  marking  out  the  path. 

It  is  found  that  the  radiant  point  is  always  in  the  same  position 
among  the  stars,  wherever  the  observer  may  be  situated,  and  that 


METEORS  AND  COMETS.  383 

it  does  not  partake  of  the  diurnal  motion  of  the  earth — that  is,  as 
the  stars  apparently  move  toward  the  west,  the  radiant  point  moves 
with  them. 

The  radiant  point  is  due  to  the  fact  that  the  meteoroids  which 
strike  the  earth  during  a  shower  are  all  moving  in  the  same  direc- 
tion. If  we  suppose  the  earth  to  be  at  rest,  and  the  actual  motion 
of  the  meteoroids  to  be  compounded  with  an  imaginary  motion 
equal  and  opposite  to  that  of  the  earth,  the  motion  of  these  imag- 
inary bodies  will  be  the  same  as  the  actual  relative  motion  of  the 
meteoroids  seen  from  the  earth.  These  relative  motions  will  all  be 
parallel  ;  hence  when  the  bodies  strike  our  atmosphere  the  paths 
described  by  them  in  their  passage  will  all  be  parallel  straight 
lines.  Now,  by  the  principles  of  spherical  trigonometry,  a  straight 
line  seen  by  an  observer  at  any  point  is  projected  as  a  great  circle 
of  the  celestial  sphere,  of  which  the  observer  supposes  himself  to  be 
the  centre.  If  we  draw  a  line  from  the  observer  parallel  to  the 
paths  of  the  meteors,  the  direction  of  that  line  will  represent  a  point 
of  the  sphere  through  which  all  the  paths  will  seem  to  pass  ;  this 
will,  therefore,  be  the  radiant  point  in  a  meteoric  shower. 

A  slightly  different  conception  of  the  problem  may  be  formed 
by  conceiving  the  plane  passing  through  the  observer  and  contain- 
ing the  path  of  the  meteor.  It  is  evident  that  the  different  planes 
formed  by  the  parallel  meteor  paths  will  all  intersect  each  other  in 
a  line  drawn  from  the  observer  parallel  to  this  path.  This  line 
will  then  intersect  the  celestial  sphere  in  the  radiant  point. 

Orbits  of  Meteoric  Showers. — From  what  has  just  been  said, 
it  will  be  seen  that  the  position  of  the  radiant  point  indicates  the 
direction  in  which  the  meteoroids  move  relatively  to  the  earth.  If 
we  also  knew  the  velocity  with  which  they  are  really  moving  in 
space,  we  could  make  allowance  for  the  motion  of  the  earth,  and 
thus  determine  the  direction  of  their  actual  motion  in  space.  It 
will  be  remembered  that,  as  just  explained,  the  apparent  or  rela- 
tive motion  is  made  up  of  two  components — the  one  the  actual 
motion  of  the  body,  the  other  the  motion  of  the  earth  taken  in  an 
opposite  direction.  We  know  the  second  of  these  components 
already  ;  and  if  we  know  the  velocity  relative  to  the  earth  and  the 
direction  as  given  by  the  radiant  point,  we  have  given  the  resultant 
and  one  component  in  magnitude  and  direction.  The  computation 
of  the  other  component  is  one  of  the  simplest  problems  in  kine- 
matics. Thus  we  shall  have  the  actual  direction  and  velocity  of 
the  meteoric  swarm  in  space.  Having  this  direction  and  velocity, 
the  orbit  of  the  swarm  around  the  sun  admits  of  being  calculated. 

Relations  of  Meteors  and  Comets. — The  velocity  of  the 
meteoroids  does  not  admit  of  being  determined  from  ob- 
servation. One  element  necessary  for  determining  the 
orbits  of  these  bodies  is,  therefore,  wanting.  In  the  case 
of  the  showers  of  1799,  1833,  and  1866,  commonly  called 
the  November  showers,  this  element  is  given  by  the  time 


384  A8TRONOM7. 

of  revolution  around  the  sun.  Since  the  showers  occur  at 
intervals  of  about  a  third  of  a  century,  it  is  highly  prob- 
able this  is  the  periodic  time  of  the  swarm  around  the  sun. 
The  periodic  time  being  known,  the  velocity  at  any  dis- 
tance from  the  sun  admits  of  calculation  from  the  theory 
of  gravitation.  Thus  we  have  all  the  data  for  determining 
the  real  orbits  of  the  group  of  meteors  around  the  sun. 
The  calculations  necessary  for  this  purpose  were  made 
by  LE  Y EERIER  and  other  astronomers  shortly  after  the 
great  shower  of  1866.  The  following  was  the  orbit  as 
given  by  LE  YERRIER  : 

Period  of  revolution 33-25  years. 

Eccentricity  of  orbit 0  •  9044. 

Least  distance  from  the  sun 0  •  9890. 

Inclination  of  orbit 165°  19'. 

Longitude  of  the  node 51°  IS'. 

Position  of  the  perihelion   (near  the  node). 

The  publication  of  this  orbit  brought  to  the  attention 
of  the  world  an  extraordinary  coincidence  which  had 
never  before  been  suspected.  In  December,  1865,  a 
faint  telescopic  comet  was  discovered  by  TEMPEL  at  Mar- 
seilles, and  afterward  by  H.  P.  TUTTLE  at  the  Naval 
O.bservatory,  Washington.  Its  orbit  was  calculated  by 
Dr.  OPPOLZER,  of  Vienna,  and  his  results  were  finally  pub- 
lished on  January  28th,  1867,  in  the  Astronomische  Nach- 
richten  ;  they  were  as  follows  : 

Period  of  revolution ...    33-18  years. 

Eccentricity  of  orbit 0  •  9054. 

Least  distance  from  the  sun 0  •  9765. 

Inclination  of  orbit 162°  42'. 

Longitude  of  the  node 51°  26'. 

Longitude  of  the  perihelion 42°  24'. 

The  publication  of  the  cometary  orbit  and  that  of  the 
orbit  of  the  meteoric  group  were  made  independently  with- 
in a  few  days  of  each  other  by  two  astronomers,  neither 
of  whom  had  any  knowledge  of  the  work  of  the  other. 
Comparing  them,  the  result  is  evident.  The  swarms  of 
meteoroids  which  cause  the  November  showers  move  in 
the  same  orbit  with  TEMPEL 's  comet. 


THE  AUGUST  MKTKORS.  385 

TEMPEL'S  comet  passed  its  perihelion  in  January, 
1866.  The  most  striking  meteoric  shower  commenced 
in  the  following  November,  and  was  repeated  during 
several  years.  It  seems,  therefore,  that  the  meteoroids 
which  produce  these  showers  follow  after  TEMPEL'S  comet, 
moving  in  the  same  orbit  with  it.  This  shows  a  curious 
relation  between  comets  and  meteors,  of  which  we  shall 
speak  more  fully  in  the  next  chapter.  When  this  fact 
was  brought  out,  the  question  naturally  arose  whether  the 
same  thing  might  not  be  true  of  other  meteoric  showers. 

Other  Showers  of  Meteors — Although  the  November 
showers  are  the  only  ones  so  brilliant  as  to  strike  the  ordi- 
nary eye,  it  has  long  been  known  that  there  are  other 
nights  of  the  year  in  which  more  shooting  stars  than  usual 
are  seen,  and  in  which  the  large  majority  radiate  from  one 
point  of  the  heavens.  This  shows  conclusively  that  they 
arise  from  swarms  of  meteoroids  moving  together  around 
the  sun. 

August  Meteors. — The  best  marked  of  these  minor 
showers  occurs  about  August  9th  or  10th  of  each  year. 
The  radiant  point  is  in  the  constellation  Perseus.  By 
watching  the  eastern  heavens  toward  midnight  on  the  9th 
or  10th  of  August  of  any  year,  it  will  be  seen  that  numer- 
ous meteors  move  from  north-east  toward  south-west,  hav- 
ing often  the  distinctive  characteristic  of  leaving  a  trail 
behind,  which,  however, vanishes  in  a  few  moments.  As- 
suming their  orbits  to  be  parabolic,  the  elements  were  cal- 
culated by  SCHIAPAKELLI,  of  Milan,  and,  on  comparing  with 
the  orbits  of  observed  comets,  it  was  found  that  these 
meteoroids  moved  in  nearly  the  same  orbit  as  the.  second 
comet  of  1862.  The  exact  period  of  this  comet  is  not 
known,  although  the  orbit  is  certainly  elliptic.  Accord- 
ing to  the  best  calculation,  it  is  124  years,  but  for  reasons 
given  in  the  next  chapter,  it  may  be  uncertain  by  ten 
years  or  more. 

There  is  one  remarkable  difference  between  the  August  and  the 
November  meteors.     The   latter,  as  we  have  seen,  appear  for  two 


386  ASTRONOMY. 

or  three  consecutive  years,  and  then  are  not  seen  again  until  about 
thirty  years  have  elapsed.  But.  the  August  meteors  are  seen  every 
year.  This  shows  that  the  stream  of  August  meteoroids  is  endless, 
every  part  of  the  orbit  being  occupied  by  them,  while  in  the  case 
of  the  November  ones  they  are  gathered  into  a  group. 

We  may  conclude  from  this  that  the  November  meteoroids  have 
not  been  permanent  members  of  our  system.  It  is  beyond  all  prob- 
ability that  a  group  comprising  countless  millions  of  such  bodies 
should  all  have  the  same  time  of  revolution.  Even  if  they  had  the 
same  time  in  the  beginning,  the  different  actions  of  the  planets  on 
different  parts  of  the  group  would  make  the  times  different.  The 
result  would  be  that,  in  the  course  of  ages,  those  which  had  the 
most  rapid  motion  would  go  further  and  further  ahead  of  the 
others  until  they  got  half  a  revolution  ahead  of  them,  and  would 
finally  overtake  those  having  the  slowest  motion.  The  swiftest  and 
slowest  one  would  then  be  in  the  position  of  two  race-horses  running 
around  a  circular  track  for  so  long  a  time  that  the  swiftest  horse 
has  made  a  complete  run  more  than  the  slowest  one  and  has  over- 
taken him  from  behind.  When  this  happens,  the  meteoroids  will 
be  scattered  all  around  the  orbit,  and  we  shall  have  a  shower  in 
November  of  every  year.  The  fact  that  has  not  yet  happened  shows 
that  they  have  been  revolving  for  only  a  limited  length  of  time, 
probably  only  a  very  few  thousand  years. 

Although  the  total  mass  of  these  bodies  is  very  small,  yet  their 
number  is  beyond  all  estimation.  Professor  NEWTON  has  estimated 
that,  taking  the  whole  earth,  about  seven  million  shooting  stars  are 
encountered  every  twenty-four  hours.  This  would  make  between 
two  and  three  thousand  million  meteoroids  which  are  thus,  as  it 
were,  destroyed  every  year.  But  the  number  which  the  earth  can 
encounter  in  a  year  is  only  an  insignificant  fraction  of  the  total 
number,  even  in  the  solar  system.  It  may  be  interesting  to  calculate 
the  ratio  of  the  space  swept  over  by  the  earth  in  the  course  of  a  year 
to  the  volume  of  the  sphere  surrounding  the  sun  and  extending  out 
to  the  orbit  of  Neptune.  We  shall  find  this  ratio  to  be  only  as  one 
to  about  three  millions  of  millions.  If  we  measure  by  the  number 
of  meteoroids  in  a  cubic  mile,  we  might  consider  them  very  thinly 
scattered.  There  is,  in  fact,  only  a  single  meteor  to  several  million 
cubic  kilometres  of  space  in  the  heavens.  Yet  the  total  number 
is  immensely  great,  because  a  globe  including  the  orbit  of  Neptune 
would  contain  millions  of  millions  of  millions  of  millions  of  cubic 
kilometres.*  If  we  reflect,  in  addition,  that  the  meteoroids  probably 

*  The  computations  leading  to  this  result  may  be  made  in  the  fol- 
lowing manner : 

I.  To  find  tfie  cubical  space  swept  through  by  the  earth  in  the  course  of 
a  year.  If  we  put  TT  for  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter,  and  p  for  the  radius  of  the  earth,  the  surface  of  a  plane  section 
of  the  earth  passing  through  its  centre  will  be  TT  p2.  Multiplying  this 
by  the  circumference  of  the  earth's  orbit,  we  shall  have  the  space  re- 
quired, which  we  readily  find  to  be  more  than  30,000  millions  of 
millions  of  kilometres.  Since,  in  sweeping  through  this  space,  the 
earth  encounters  alxmt  2500  millions  of  meteoroids,  it  follows  that 


THE  ZODIACAL  LIGHT.  387 

weigh  but  a  few  grains  each,  we  shall  see  how  it  is  that  they  are  en- 
tirely invisible  to  vision,  even  with  powerful  telescopes. 

The  Zodiacal  Light.  —  If  we  observe  the  'western  sky 
during  the  winter  or  spring  months,  about  the  end  of  the 
evening  twilight,  we  shall  see  a  stream  of  faint  light,  a 
little  like  the  Milky  Way,  rising  obliquely  from  the  west, 
and  directed  along  the  ecliptic  toward  a  point  south-west 
from  the  zenith.  This  is  called  the  zodiacal  light.  It 
may  also  be  seen  in  the  east  before  daylight  in  the  morn- 
ing during  the  autumn  months,  and  has  sometimes  been 
traced  all  the  way  across  the  heavens.  Its  origin  is  still 
involved  in  obscurity,  but  it  seems  probable  that  it  arises 
from  an  extremely  thin  cloud  either  of  meteoroids  or  of 
semi-gaseous  matter  like  that  composing  the  tail  of  a 
comet,  spread  all  around  the  sun  inside  the  earth's  orbit. 
The  researches  of  Professor  A.  W.  WRIGHT  show  that  its 
spectrum  is  probably  that  of  reflected  sunlight,  a  result 
which  gives  color  to  the  theory  that  it  arises  from  a  cloud 
of  meteoroids  revolving  round  the  sun. 

there  is  only  one  meteoroid  to  more  than  ten  millions  of  cubic  kil- 
ometres. 

II.  To  find  the  ratio  of  the  sphere  of  space  within  the  orbit  of  Neptune  to 
the  space  swept  through  by  the  earth  in  a  year.  Let  us  put  r  for  the  dis- 
tance of  the  earth  from  the  sun.  Then  the  distance  of  Neptune  may 
be  taken  as  30  r,  and  this  will  be  the  radius  of  the  sphere.  The  cir- 
cumference of  the  earth's  orbit  will  than  be  2  TT  ?%  and  the  space  swept 
over  will  be  2  ;r2  r  p2.  The  sphere  of  Neptune  will  be 

*  TT  303  ?-3  =  36,000  TT  r3,  nearly. 
The  ratio  of  the  two  spaces  will  be 

I'  =  6,000  4,  nearly. 

* 


The  ratio  —    is  more  than  23,000,  showing  the  required  ratio   to  be 

P 

about  three  millions  of  millions.     The  total  number  of  scattered  mete- 
oroids is  therefore  to  be  reckoned  by  millions  of  millions  of  millions. 


CHAPTER    XIII. 

COMETS. 
§   1.    ASPECT  OP  COMETS. 

COMETS  are  distinguished  from  the  planets  both  by  their 
aspects  and  their  motions.  They  come  into  view  without 
anything  to  herald  their  approach,  continue  in  sight  for  a 
few  weeks  or  months,  and  then  gradually  vanish  in  the 
distance.  They  are  commonly  considered  as  composed  of 
three  parts,  the  nucleus,  the  coma  (or  hair),  and  the  tail. 

The  nucleus  of  a  comet  is,  to  the  naked  eye,  a  point  of 
light  resembling  a  star  or  planet.  Viewed  in  a  telescope, 
it  generally  has  a  small  disk,  but  shades  off  so  gradually 
that  it  is  difficult  to  estimate  its  magnitude.  In  large 
comets,  it  is  sometimes  several  hundred  miles  in  diameter, 
but  never  approaches  the  size  of  one  of  the  larger  planets. 

The  nucleus  is  always  surrounded  by  a  mass  of  foggy 
light,  which  is  called  the  coma.  To  the  naked  eye,  the 
nucleus  and  coma  together  look  like  a  star  seen  through  a 
mass  of  thin  fog,  which  surrounds  it  with  a  sort  of  halo. 
The  coma  is  brightest  near  the  nucleus,  so  that  it  is  hardly 
possible  to  tell  where  the  nucleus  ends  and  where  the 
coma  begins.  It  shades  off  in  every  direction  so  gradually 
that  no  definite  boundaries  can  be  fixed  to  it.  The 
nucleus  and  coma  together  are  generally  called  the  head 
of  the  comet. 

The  tail  of  the  comet  is  simply  a  continuation  of  the 
coma  extending  out  to  a  great  distance,  and  always  di- 
rected away  from  the  sun.  It  has  the  appearance  of  a 
stream  of  milky  light,  which  grows  fainter  and  broader 


ASPECT  OF  COMETS.  389 

as  it  recedes  from  the  head.  Like  the  coma,  it  shades  off 
so  gradually  that  it  is  impossible  to  fix  any  boundaries  to 
it.  The  length  of  the  tail  varies  from  2°  or  3°  to  90°  or 
more.  Generally  the  more  brilliant  the  head  of  the  comet, 
the  longer  and  brighter  is  the  tail.  It  is  also  often  brighter 
and  more  sharply  defined  at  one  edge  than  at  the  other. 

The  above  description  applies  to  comets  which  can  be 
plainly  seen  by  the  naked  eye.  After  astronomers  began 
to  sweep  the  heavens  carefully  with  telescopes,  it  was 
found  that  many  comets  came  into  sight  which  would 
entirely  escape  the  unaided  vision.  These  are  called  tel- 
escopic comets.  Sometimes  six  or  more  of  such  comets  are 
discovered  in  a  single  year,  while  one  of  the  brighter  class 
may  not  be  seen  for  ten  years  or  more. 


FlG.  101.— TELESCOPIC  COMET  WITH-  FlG.  102.— TELESCOPIC  COMET 
OUT  A  NUCLEUS.  WITH  A  NUCLEUS. 

When  comets  are  studied  with  a  telescope,  it  is  found 
that  they  are  subject  to  extraordinary  changes  of  structure. 
To  understand  these  changes,  we  must  begin  by  saying  that 
comets  do  not,  like  the  planets,  revolve  around  the  sun  in 
nearly  circular  orbits,  but  always  in  orbits  so  elongated 
that  the  comet  is  visible  in  only  a  very  small  part  of  its 
course.  When  one  of  these  objects  is  first  seen,  it  is  gen- 
erally approaching  the  sun  from  the  celestial  spaces. 
At  this  time  it  is  nearly  always  devoid  of  a  tail,  and  some- 
times of  a  nucleus,  presenting  the  aspect  of  a  thin  patch 
of  cloudy  light,  which  may  or  may  not  have  a  nucleus  in 


390  ASTRONOMY. 

its  centre.  As  it  approaches  the  sun,  it  is  generally  seen 
to  grow  brighter  at  some  #one  point,  and  there  a  nucleus 
gradually  forms,  being,  at  first,  so  faint  that  it  can  scarcely 
be  distinguished  from  the  surrounding  nebulosity.  The 
latter  is  generally  more  extended  in  the  direction  of  the 
sun,  thus  sometimes  giving  rise  to  the  erroneous  impres- 
sion of  a  tail  turned  toward  the  sun.  Continuing  the 
watch,  the  true  tail,  if  formed  at  all,  is  found  to  begin 
very  gradually.  At  first  so  small  and  faint  as  to  be  almost 
invisible,  it  grows  longer  and  brighter  every  day,  as  long 
as  the  comet  continues  to  approach  the  sun. 

§   2.    THE  VAPOROUS  ENVELOPES. 

If  a  comet  is  very  small,  it  may  undergo  no  changes  of 
aspect,  except  those  just  described.  If  it  is  an  unusually 
bright  one,  the  next  object  noticed  by  telescopic  examina- 
tion will  be  a  bow  surrounding  the  nucleus  on  the  side 
toward  the  sun.  This  bow  will  gradually  rise  up  and 
spread  out  on  all  sides,  finally  assuming  the  form  of  a 
semicircle  having  the  nucleus  in  its  centre,  or,  to  speak 
with  more  precision,  the  form  of  a  parabola  having  the 
nucleus  near  its  focus.  The  two  ends  of  this  parabola 
will  extend  out  further  and  further  so  as  to  form  a  part 
of  the  tail,  and  finally  be  lost  in  it.  Continuing  the 
watch,  other  bows  will  be  found  to  form  around  the  nu- 
cleus, all  slowly  rising  from  it  like  clouds  of  vapor. 
These  distinct  vaporous  masses  are  called  the  envelopes  : 
they  shade  off  gradually  into  the  coma  so  as  to  be  with 
difficulty  distinguished  from  it,  and  indeed  may  be  con- 
sidered as  part  of  it.  The  inner  envelope  is  sometimes 
connected  with  the  nucleus  by  one  or  more  fan -shaped 
appendages,  the  centre  of  the  fan  being  in  the  nucleus, 
and  the  envelope  forming  its  round  edge.  This  appear- 
ance is  apparently  caused  by  masses  of  vapor  streaming 
up  from  that  side  of  the  nucleus  nearest  the  sun,  and  grad- 
ually spreading  around  the  comet  on  each  side.  The 


ENVELOPES  OF  COMETS.  391 

form  of  a  bow  is  not  the  real  form  of  the  envelopes,  but 
only  the  apparent  one  in  which  we  see  them  projected 
against  the  background  of  the  sky.  Their  true  form  is 
similar  to  that  of  a  paraboloid  of  revolution,  surrounding 
the  nucleus  on  all  sides,  except  that  turned  from  the  sun. 
It  is,  therefore,  a  surface  and  not  a  line.  Perhaps  its  form 
can  be  best  imagined  by  supposing  the  sun  to  be  directly 
above  the  comet,  and  a  fountain,  throwing  a  liquid  hori- 
zontally on  all  sides,  to  be  built  upon  that  part  of  the 
cornet  which  is  uppermost.  Such  a  fountain  would  throw 
its  water  in  the  form  of  a  sheet,  falling  on  all  sides  of  the 
cometic  nucleus,  but  not  touching  it.  Two  or  three  vapor 
surfaces  of  this  kind  are  sometimes  seen  around  the  comet, 
the  outer  one  enclosing  each  of  the  inner  ones,  but  no  two 
touching  each  other. 


FlG.  103. — FORMATION  OP  ENVELOPES. 

To  give  a  clear  conception  of  the  formation  and  motion  of  the 
envelopes,  we  present  two  figures.  The  first  of  these  shows  the  ap- 
pearance of  the  envelopes  in  four  successive  stages  of  their  course, 
and  may  be  regarded  as  sections  of  the  real  umbrella-shaped  sur- 
faces which  they  form.  In  all  these  figures,  the  sun  is  supposed  to 
be  above  the  comet  in  the  figure,  and  the  tail  of  the  comet  to  be 
directed  downward.  In  a  the  sheet  of  vapor  has  just  begun  to 
rise.  In  5  it  is  risen  and  expanded  yet  further.  In  c  it  has  begun 
to  move  away  and  pass  around  the  comet  on  all  sides.  Finally, 
in  d  this  last  motion  has  gone  so  far  that  the  higher  portions 
have  nearly  disappeared,  the  larger  part  of  the  matter  having 
moved  away  toward  the  tail.  Before  the  stage  c  is  reached,  a 
second  envelope  will  commonly  begin  to  rise  as  at  «,  so  that  two 
or  three  may  be  visible  at  the  same  time,  enclosed  within  each 
other. 

In  the  next  figure  the   actual  motion   of   the  matter  compos- 


393  ASTRONOMY. 

ing  the  envelopes  is  shown  by  the  courses  of  the  several  dotted 
lines.  This  motion,  it  will  be  seen,  is  not  very  unlike  that  of 
water  thrown  up  from  a  founfain  on  the  part  of  the  nucleus 
nearest  the  sun  and  then  falling  down  on  all  sides.  The  point  in 
which  the  motion  of  the  cometic  matter  differs  from  that  of  the 
fountain  is  that,  instead  of  being  thrown  in  continuous  streams, 
the  action  is  intermittent,  the  fountain  throwing  up  successive 
sheets  of  matter  instead  of  continuous  streams. 

From  the  gradual  expansion  of  these  envelopes  around  the  head 
of  the  comet  and  the  continual  formation  of  new  ones  in  the  im- 
mediate neighborhood  of  the  nucleus,  they  would  seem  to  be  due 
to  a  process  of  evaporation  going  on  from  the  surface  of  the  latter. 
Each  layer  of  vapor  thus  formed  rises  up  and  spreads  out  con- 
tinually until  the  part  next  the  sun  attains  a  certain  maximum 
height.  Then  it  gradually  moves  away  from  the  sun,  keeping  its 
distance  from  the  comet,  at  least  until  it  passes  the  latter  on  every 
side,  and  continues  onward  to  form  the  tail. 


FlG.  104. — FORMATION  OF  COMET'S  TAIL. 

These  phenomena  were  fully  observed  in  the  great  comet  of 
1858,  the  observations  of  which  were  carefully  collected  by  the 
late  Professor  BOND,  of  Cambridge.  The  envelopes  of  this  comet 
were  first  noticed  on  September  20th,  when  the  outer  one  was  16" 
above  the  nucleus  and  the  inner  one  3".  The  outer  one  disap- 
peared on  September  30th  at  a  height  of  about  1'.  In  the  mean 
while,  however,  a  third  had  appeared,  the  second  having  gradually 
expanded  so  as  to  take  the  place  of  the  first.  Seven  successive 
envelopes  in  all  were  seen  to  rise  from  this  comet,  the  last  one  com- 
mencing on  October  20th,  when  all  the  others  had  been  dissipated. 
The  rate  at  which  the  envelopes  ascended  was  generally  from  50  to 
60  kilometres  per  hour,  the  ordinary  speed  of  a  railway-train. 

The  first  one  rose  to  a  height  of  about  30,000  kilometres,  but  it 
was  finally  dissipated.  But  the  successive  ones  disappeared  at  a 
lower  and  lower  elevation,  the  sixth  being  lost  sight  of  at  a  height 
of  about  10,000  kilometres. 


SPECTRA   OF  COMETS. 


393 


In  the  great  comet  of  1861,  eleven  envelopes  were  seen  between 
July  3d,  when  portions  of  three  were  in  sight,  and  the  19th  of 
the  same  month,  a  new  one  rising  at  regular  intervals  of  every  sec- 
ond day.  Their  evolution  and  dissipation  were  accomplished  with 
much  greater  rapidity  than  in  the  case  of  the  great  comet  of  1858, 
an  envelope  requiring  but  two  or  three  days  instead  of  two  or  three 
weeks  to  pass  through  all  its  phases. 


§   3.    THE  PHYSICAL   CONSTITUTION  OF  COMETS. 

To  tell  exactly  what  a  comet  is,  we  should  be  able  to 
show  how  all  the  phenomena  it  presents  would  follow  from 
the  properties  of  matter,  as  we  learn  them  at  the  surface 
of  the  earth.  This,  however,  no  one  has  been  able  to  do, 
many  of  the  phenomena  being  such  as  we  should  not  ex- 
pect from  the  known  constitution  of  matter.  All  we  can 
do,  therefore,  is  to  present  the  principal  characteristics  of 
comets,  as  shown  by  observation,  and  to  explain  what  is 
wanting  to  reconcile  these  characteristics  with  the  known 
properties  of  matter. 

In  the  first  place,  all  comets  which  have  been  examined 
with  the  spectroscope  show  a  spectrum  composed,  in  part 
at  least,  of  bright  lines  or  bands.  These  lines  have  been 
supposed  to  be  identified  with  those  of  carbon ;  but 
although  the  similarity  of  aspect  is  very  striking,  the  iden- 
tity cannot  be  regarded  as  proven. 


FlG.  105. — SPECTRA  OP  OLEFIANT  GAS  AND  OP  A  COMET. 

In  the  annexed  figure  the  upper  spectrum,  A,  is  that  of  carbon 
taken  in  olefiant  gas,  and  the  lower  one,  B,  that  of  a  comet.  These 
spectra  interpreted  in  the  usual  way  would  indicate,  firstly,  thai 
the  comet  is  gaseous  ;  secondly,  that  the  gases  which  compose  it 
are  so  hot  as  to  shine  by  their  own  light.  But  we  cannot  admit 


394  ASTRONOMY. 

these  interpretations  without  bringing  in  some  additional  theory. 
A  mass  of  gas  surrounding  so  minute  a  body  as  the  nucleus  of  a 
telescopic  comet  would  expand  into  space  by  virtue  of  its  own 
elasticity  unless  it  were  exceedingly  rare.  Moreover,  if  it  were 
incandescent,  it  would  speedily  cool  off  so  as  to  be  no  longer  self- 
luminous.  We  must,  therefore,  propose  some  theory  to  account 
for  the  continuation  of  the  luminosity  through  many  centuries, 
such  as  electric  activity  or  phosphorescence.  But  without  further 
proof  of  action  of  these  causes  we  cannot  accept  their  reality.  We 
are,  therefore,  unable  to  say  with  certainty  how  the  light  in  the 
spectrum  of  comets  which  produces  the  bright  lines  has  its  origin. 

In  the  last  chapter  it  was  shown  that  swarms  of  minute 
particles  called  meteoroids  follow  certain  comets  in  their 
orbits.  This  is  no  doubt  true  of  all  comets.  We  can  only 
regard  these  meteoroids  as  fragments  or  debris  of  the 
comet.  The  latter  has  therefore  been  considered  by  Pro- 
fessor NEWTON  as  made  up  entirely  of  meteoroids  or  small 
detached  masses  of  matter.  These  masses  are  so  small  and 
so  numerous  that  they  look  like  a  cloud,  and  the  light 
which  they  reflect  to  our  eyes  has  the  milky  appearance 
peculiar  to  a  comet.  On  this  theory  a  telescopic  comet 
which  has  no  nucleus  is  simply  a  cloud  of  these  minute 
bodies.  The  nucleus  of  the  brighter  comets  may  either 
be  a  more  condensed  mass  of  such  bodies  or  it  may  be  a 
solid  or  liquid  body  itself. 

If  the  reader  has  any  difficulty  in  reconciling  this  theory 
of  detached  particles  with  the  view  already  presented, 
that  the  envelopes  from  which  the  tail  of  the  cornet  is 
formed  consist  of  layers  of  vapor,  he  must  remember  that 
vaporous  masses,  such  as  clouds,  fog,  and  smoke,  are 
really  composed  of  minute  separate  particles  of  water  or 
carbon. 

Formation  of  the  Comet's  Tail. — The  tail  of  the  comet 
is  not  a  permanent  appendage,  but  is  composed  of  the 
masses  of  vapor  which  we  have  already  described  as  as- 
cending from  the  nucleus,  and  afterward  moving  away 
from  the  sun.  The  tail  which  we  see  on  one  evening  is 
not  absolutely  the  same  we  saw  the  evening  before,  a 


MOTIONS  OF  COMETS.  395 

portion  of  the  latter  having  been  dissipated,  while  new 
matter  lias  taken  its  place,  as  with  the  stream  of  smoke  from 
a  steamship.  The  motion  of  the  vaporous  matter  which 
forms  the  tail  being  always  away  from  the  sun,  there 
seems  to  be  a  repulsive  force  exerted  by  the  sun  upon  it. 
The  form  of  the  comet's  tail,  on  the  supposition  that  it  is 
composed  of  matter  thus  driven  away  from  the  sun  with 
a  uniformly  accelerated  velocity,  has  been  several  times 
investigated,  and  found  to  represent  the  observed  form  of 
the  tail  so  nearly  as  to  leave  little  doubt  of  its  correctness. 
We  may,  therefore,  regard  it  as  an  observed  fact  that  the 
vapor  which  rises  from  the  nucleus  of  the  comet  is  repelled 
by  the  sun  instead  of  being  attracted  toward  it,  as  larger 
masses  of  matter  are. 

No  adequate  explanation  of  this  repulsive  force  has  ever  been 
given.  It  has,  indeed,  been  suggested  that  it  is  electrical  in  its 
character,  but  no  one  has  yet  proven  experimentally  that  the  attrac- 
tion exerted  by  the  sun  upon  terrestrial  bodies  is  influenced  by  their 
electrical  state.  If  this  were  done,  we  should  have  a  key  to  one  of 
the  most  difficult  problems  connected  with  the  constitution  of 
comets.  As  the  case  now  stands,  the  repulsion  of  the  sun  upon  the 
comet's  tail  is  to  be  regarded  as  a  well-ascertained  and  entirely 
isolated  fact  which  has  no  known  counterpart  in  any  other  observed 
fact  of  nature. 

In  view  of  the  difficulties  we  find  in  explaining  the  phenomena  of 
comets  by  principles  based  upon  our  terrestrial  chemistry  and 
physics,  the  question  will  arise  whether  the  matter  which  composes 
these  bodies  may  not  be  of  a  constitution  entirely  different  from 
that  of  any  matter  we  are  acquainted  with  at  the  earth's  surface. 
If  this  were  so,  it  would  be  impossible  to  give  a  complete  explanation 
of  comets  until  we  know  what  forms  matter  might  possibly  assume 
different  from  those  we  find  it  to  have  assumed  in  our  labora- 
tories. This  is  a  question  which  we  merely  suggest  without 
attempting  to  speculate  upon  it.  It  can  be  answered  only  by  ex- 
perimental researches  in  chemistry  and  physics. 


§  4.    MOTIONS  OP  COMETS. 

Previous  to  the  time  of  NEWTON,  no  certain  knowledge 
respecting  the  actual  motions  of  comets  in  the  heavens 
had  been  acquired,  except  that  they  did  not  move  around 


396  ASTRONOMY. 

the  sun  like  the  planets.  When  NEWTON  investigated  the 
mathematical  results  of  the  theory  of  gravitation,  he  found 
that  a  body  moving  under  the  attraction  of  the  sun  might 
describe  either  of  the  three  conic  sections,  the  ellipse,  par- 
abola, or  hyperbola.  Bodies  moving  in  an  ellipse,  as  the 
planets,  would  complete  their  orbits  at  regular  intervals 
of  time,  according  to  laws  already  laid  down.  But  if  the 
body  moved  in  a  parabola  or  a  hyperbola,  it  would  never 
return  to  the  sun  after  once  passing  it,  but  would  move  off 


PlQ.    106.— ELLIPTIC  AND  PAKABOLIC  ORBITS. 

to  infinity.  It  was,  therefore,  very  natural  to  conclude 
that  comets  might  be  bodies  which  resemble  the  planets  in 
moving  under  the  sun's  attraction,  but  which,  instead  of 
describing  an  ellipse  in  regular  periods,  like  the  planets, 
move  in  parabolic  or  hyperbolic  orbits,  and  therefore 
only  approach  the  sun  a  single  time  during  their  whole 
existence. 

This  theory  is  now  known  to  be  essentially  true  for 


ORBITS  OF  COMETS.  397 

most  of  the  observed  comets.  A  few  are  indeed  found  to 
be  revolving  around  the  sun  in  elliptic  orbits,  which  differ 
from  those  of  the  planets  only  in  being  much  more  eccen- 
tric. But  the  greater  number  which  have  been  observed 
have  receded  from  the  sun  in  orbits  which  we  are  unable 
to  distinguish  from  parabolas,  though  it  is  possible  they 
may  be  extremely  elongated  ellipses.  Comets  are  there- 
fore divided  with  respect  to  their  motions  into  two  classes  : 
(1)  periodic  comets,  which  are  known  to  move  in  elliptic 
orbits,  and  to  return  to  the  sun  at  fixed  intervals  ;  and  (2) 
parabolic  comets,  apparently  moving  in  parabolas,  never 
to  return. 

The  first  discovery  of  the  periodicity  of  a  comet  was 
made  by  HALLEY  in  connection  with  the  great  comet  of 
1082.  Examining  the  records  of  observations,  he  found 
that  a  comet  moving  in  nearly  the  same  orbit  with  that  of 
1682  had  been  seen  in  1607,  and  still  another  in  1531. 
He  was  therefore  led  to  the  conclusion  that  these  three 
cornets  were  really  one  and  the  same  object,  returning  to 
the  sun  at  intervals  of  about  75  or  76  years.  He  there- 
fore predicted  that  it  would  appear  again  about  the  year 
1758.  But  such  a  prediction  might  be  a  year  or  more  in 
error,  owing  to  the  effect  of  the  attraction  of  the  planets 
upon  the  comet.  In  the  mean  time  the  methods  of  calcu- 
lating the  attraction  of  the  planets  were  so  far  improved 
that  it  became  possible  to  make  a  more  accurate  predic- 
tion. As  the  year  1759  approached,  the  necessary  com- 
putations were  made  by  the  great  French  geometer  CLAI- 
KAUT,  who  assigned  April  13th,  1759,  as  the  day  on  which 
the  comet  would  pass  its  perihelion.  This  prediction 
was  remarkably  correct.  The  comet  was  first  seen  on 
Christmas-day,  1758,  and  passed  its  perihelion  March 
12th,  1759,  only  one  month  before  the  predicted  time. 
The  comet  returned  again  in  1835,  within  three  days  of 
the  moment  predicted  by  DE  PONTECOULANT,  the  most 
successful  calculator.  The  next  return  will  probably  take 


398  ASTRONOMY, 

place  in  1911  or  1912,  the  exact  time  being  still  unknown, 

because  the  necessary  computations  have  not   yet  been 

made. 

We  give  a  figure  showing  the  position  of  the  orbit  of 

HALLEY'S  comet  relative  to  the  orbits  of  the  four  outer 

planets.  It  attain- 
ed its  greatest  dis- 
tance from  the  sun, 
far  beyond  the  or- 
bit of  Neptune, 
about  the  year  1 8  73 , 
and  tli en  c o m- 
menced  its  return 
journey.  The  fig- 
ure shows  the  prob- 
able position  of  the 
comet  in  1874.  It 
was  then  far  be- 

FIG.  107.-OBB,r  OF  HALLEY'S  COMET.        y°nd  th°   reac1'  °^ 

the  most  powerful 

telescope,  but  its  distance  and  direction  admit  of  being 
calculated  with  so  much  precision  that  a  telescope  could 
be  pointed  at  it  at  any  required  moment. 

We  have  already  stated  that  great  numbers  of  comets, 
too  faint  to  be  seen  by  the  naked  eye,  are  discovered  by 
telescopes.  A  considerable  number  of  these  telescopic 
comets  have  been  found  to  be  periodic.  In  most  cases, 
the  period  is  many  centuries  in  length,  so  that  the  comets 
have  only  been  noticed  at  a  single  visit.  Eight  or 
nine,  however,  have  been  found  to  be  of  a  period  so  short 
that  they  have  been  observed  at  two  or  more  returns. 

We  present  a  table  of  such  of  the  periodic  comets  as 
have  been  actually  observed  at  two  or  more  returns.  A 
number  of  others  are  known  to  be  periodic,  but  have  been 
observed  only  on  a  single  visit  to  our  system. 


ORBITS  OF  COMETS. 


399 


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400  ASTRONOMY. 

Theory  of  Cometary  Orbits.  —  There  is  a  property  of  all  or- 
bits of  bodies  around  the  sun.  an  understanding  of  which  will 
enable  us  to  form  a  clear  idea  of  some  causes  which  affect  the 
motion  of  comets.  It  may  be  expressed  in  the  following  theorem  : 

The  mean  distance  of  a  tody  from  the  sun,  or  the  major  axis  of  the 
ellipse  in  which  it  revolves,  depends  only  upon  the  velocity  of  the 
body  at  a  given  distance  from  the  sun,  and  may  be  found  by  the 
formula, 


in  which  r  is  the  distance  from  the  sun,  v  the  velocity  with  which 
the  body  is  moving,  and  ft  a  constant  proportional  to  the  mass  of 
the  sun  and  depending  on  the  units  of  time  and  length  we  adopt. 

To  understand  this  formula,  let  us  imagine  ourselves  in  the  celes- 
tial spaces,  with  no  planets  in  our  neighborhood.  Suppose  we  have 
a  great  number  of  balls  and  shoot  them  out  with  the  same  velocity, 
but  in  different  directions,  so  that  they  will  describe  orbits  around  the 
sun.  Then  the  bodies  will  all  describe  different  orbits,  owing  to 
the  different  directions  in  which  we  threw  them,  but  these  orbits 
will  all  possess  the  remarkable  property  of  having  equal  major 
axes,  and  therefore  equal  mean  distances  from  the  sun.  Since,  by 
KEPLER'S  third  law,  the  periodic  time  depends  only  upon  the 
mean  distance,  it  follows  that  the  bodies  will  have  the  same  time 
of  revolution  around  the  sun.  Consequently,  if  we  wait  patiently 
at  the  point  of  projection,  they  will  all  make  a  revolution  in  the 
same  time,  and  will  all  come  back  again  at  the  same  moment,  each 
one  coming  from  a  direction  the  opposite  of  that  in  which  it  was 
thrown. 

In  the  above  formula  the  major  axis  is  given  by  a  fraction,  having 
the  expression  2  /*  —  r  v*  for  its  denominator  ;  it  follows  that  if  the 

square  of  the  velocity  is  almost  equal  to  —  ,  the  value  of  a  will 

become  very  great,  because  the  denominator  of  the  fraction  will  be 
very  small.  If  the  velocity  is  such  that  2  p  —  rv*  is  zero,  the  mean 
distance  will  become  infinite.  Hence,  in  this  case  the  body  will 
fly  off  to  an  infinite  distance  from  the  sun  and  never  return. 
Much  less  will  it  return  if  the  velocity  is  still  greater.  Such  a 
velocity  will  make  the  value  of  a  algebraically  negative  and  will 
correspond  to  the  hyperbola. 

If  we  take  one  kilometre  per  second  as  the  unit  of  velocity,  and 
the  mean  distance  of  the  earth  from  the  sun  as  the  unit  of  distance, 
the  value  of  j«  will  be  represented  by  the  number  875,  so  that  the 

875  r 
formula  for  a  will  be  a  =  -  —  —  —  —  -.     From  this  equation,  we  may 

calculate  what  velocity  a  body  moving  around  the  sun  must  have 
at  any  given  distance  r,  in  order  that  it  may  move  in  a  parabolic 
orbit  —  that  is,  that  the  denominator  of  the  fraction  shall  vanish. 

This  condition  will  give  «2  =   -    —  .     At  the  distance  of  the  earth 


ORIGIN  OF  COMETS.  401 

from  the  sun  we  have  r  =  1,  so  that,  at  that  distance,  v  will  be  the 
square  root  of  1750,  or  nearly  42  kilometres  per  second.  The  fur- 
ther we  get  out  from  the  sun,  the  less  it  will  be  ;  and  we  may  remark, 
as  an  interesting  theorem,  that  whenever  the  comet  is  at  the  dis- 
tance of  one  of  the  planetary  orbits,  its  velocity  must  be  equal  to 
that  of  the  planet  multiplied  by  the  square  root  of  2,  or  1  -414,  etc. 
Hence,  if  the  velocity  of  any  planet  were  suddenly  increased  by  a 
little  more  than  -^  of  its  amount,  its  orbit  would  be  changed  into 
a  parabola,  and  it  would  fly  away  from  the  sun,  never  to  return. 

It  follows  from  all  this  that  if  the  astronomer,  by  observing  the 
course  of  a  comet  along  its  orbit,  can  determine  its  exact  velocity 
from  point  to  point,  he  can  thence  calculate  its  mean  distance  from 
the  sun  and  its  periodic  time.  But  it  is  found  that  the  velocity  of 
a  large  majority  of  comets  is  so  nearly  equal  to  that  required  for 
motion  in  a  parabola,  that  the  difference  eludes  observation.  It  is 
hence  concluded  that  most  comets  move  nearly  in  parabolas,  and 
will  either  never  return  at  all  or,  at  best,  not  until  after  the  lapse  of 
many  centuries. 


§   5.    ORIGIN  OP  COMETS. 

All  that  we  know  of  comets  seems  to  indicate  that  they 
did  not  originally  belong  to  our  system,  but  became  mem- 
bers of  it  through  the  disturbing  forces  of  the  planets. 
From  what  was  said  in  the  last  section,  it  will  be  seen  that 
if  a  comet  is  moving  in  a  parabolic  orbit,  and  its  velocity 
is  diminished  at  any  point  by  ever  so  small  an  amount,  its 
orbit  will  be  changed  into  an  ellipse  ;  for  in  order  that  the 
orbit  may  be  parabolic,  the  quantity  2  \i— r  v*  must  remain 
exactly  zero.  But  if  we  then  diminish  v  by  the  smallest 
amount,  this  expression  will  become  finite  and  positive, 
and  a  will  no  longer  be  infinite.  Now,  the  attraction  of 
a  planet  may  have  either  of  two  opposite  effects  ;  it  may 
either  increase  or  diminish  the  velocity  of  the  comet. 
Hence  if  the  latter  be  moving  in  a  parabolic  orbit,  the  at- 
traction of  a  planet  might  either  throw  it  out  into  a  hyper- 
bolic orbit,  so  that  it  would  never  again  return  to  the  sun, 
but  wander  forever  through  the  celestial  spaces,  or  it 
might  change  its  orbit  into  a  more  or  less  elongated  ellipse. 

Suppose  CD  to  represent  a  small  portion  of  the  orbit 
of  the  planet  and  A  B  a  small  portion  of  the  orbit  of  a 
comet  passing  near  it.  Suppose  also  that  the  comet  passes 


402 


ASTRONOMY. 


a  little  in  front  of  the  planet,  and  that  the  simultaneous 
positions  of  the  two  bodies  a/e  represented  by  the  corre- 
sponding letters  of  the  alphabet,  #,  5,  <?,  d,  etc. ;  the  shortest 
distance  of  the  two  bodies  will  be  the  line  c  c,  and  it  is 
then  that  the  attraction  will  be  the  most  powerful. 
Between  c  c  and  d  d  the  planet  will  attract  the  comet  almost 
directly  backward.  It  follows  then  that  if  a  comet  pass 
the  planet  in  the  way  here  represented,  its  velocity  will  be 
retarded  by  the  attraction  of  the  latter.  If  therefore  it  be 
a  parabolic  comet,  the  orbit  will  be  changed  into  an 
ellipse.  The  nearer  it  passes  to  the  planet,  the  greater 
will  be  the  change,  so  long  as  it  passes  in  front  of  it.  If 

it  passes  behind,  the 
reverse  effect  will 
follow,  and  the  mo- 
tion will  be  accele- 
rated. The  orbit  will 
then  be  changed  into 
a  hyperbola.  The  or- 
bit finally  described 
after  the  comet  leaves 
our  system  will  de- 
pend upon  whether 
its  velocity  is  accele- 
rated or  retarded  by 
the  combined  attraction  of  all  the  planets. 

All  the  studies  which  have  been  made  of  comets  seem 
to  show  that  they  originally  moved  in  parabolic  orbits,  and 
were  brought  into  elliptic  orbits  in  this  way  by  the  attrac- 
tion of  some  planet.  The  planet  which  has  thus  brought 
in  the  greatest  number  is  no  doubt  Jupiter.  In  fact,  the 
orbits  of  several  of  the  periodic  comets  pass  very  near  to 
that  planet.  It  might  seem  that  these  orbits  ought  almost 
to  intersect  that  of  the  planet  which  changed  them.  This 
would  be  true  at  first,  but  owing  to  the  constant  change  in 
the  position  of  the  cometary  orbit,  produced  by  the  at- 
traction of  the  planets,  the  orbits  would  gradually  move 


FlG.    108. — ATTRACTION   OF  PLANET  ON 
COMET. 


ORIGIN  OF  COMETS.  403 

away  from  each  other,  so  that  in  time  there  might  be  no 
approach  whatever  of  the  planet  to  the  comet. 

A  remarkable  case  of  this  sort  was  afforded  by  a  comet 
discovered  in  June,  1770.  It  was  observed  in  all  nearly 
four  months,  and  was  for  some  time  visible  to  the  naked 
eye.  On  calculating  its  orbit  from  all  the  observations, 
the  astronomers  were  astonished  to  find  it  to  be  an  ellipse 
with  a  period  of  only  five  or  six  years.  It  ought  therefore 
to  have  appeared  again  in  1776  or  1777,  and  should  have 
returned  to  its  perihelion  twenty  times  before  now,  and 
should  also  have  been  visible  at  returns  previous  to  that  at 
which  it  was  first  seen.  But  not  only  was  it  never  seen 
before,  but  it  has  never  been  seen  since  !  The  reason  of 
its  disappearance  from  view  was  brought  to  light  on  cal- 
culating its  motions  after  its  first  discovery.  At  its  re- 
turn in  1776,  the  earth  was  not  in  the  right  part  of  its 
orbit  for  seeing  it.  On  passing  out  to  its  aphelion  again, 
about  the  beginning  of  1779,  it  encountered  the  planet 
Jupiter,  and  approached  so  near  it  that  it  was  impossible 
to  determine  on  which  side  it  passed.  This  approach,  it 
will  be  remembered,  could  not  be  observed,  because  the 
comet  was  entirely  out  of  sight,  but  it  was  calculated  with 
absolute  certainty  from  the  theory  of  the  comet's  motion. 
The  attraction  of  Jupiter,  therefore,  threw  it  into  some 
orbit  so  entirely  different  that  it  has  never  been  seen  since. 

It  is  also  highly  probable  that  the  comet  had  just  been 
brought  in  by  the  attraction  of  Jupiter  on  the  very  revo- 
lution in  which  it  was  first  observed.  Its  history  is  this  : 
Approaching  the  sun  from  the  stellar  spaces,  probably  for 
the  first  time,  it  passed  so  near  Jupiter  in  1767  that  its  or- 
bit was  changed  to  an  ellipse  of  short  period.  It  made 
two  complete  revolutions  around  the  sun,  and  in  1779 
again  met  the  planet  near  the  same  place  it  had  met  him 
before.  The  orbit  was  again  altered  so  much  that  no  tel- 
escope has  found  the  comet  since.  No  other  case  so  re- 
markable as  this  has  ever  been  noticed. 

Not  only  are  new  comets  occasionally  brought  in  from 


404  ASTRONOMY. 

the  stellar  spaces,  but  old  ones  may,  as  it  were,  fade  away 
and  die.  A  case  of  this  sortjs  afforded  by  BIELA'S  comet, 
which  has  not  been  seen  since  1852,  and  seems  to  have  en- 
tirely disappeared  from  the  heavens.  Its  history  is  so  in 
structive  that  we  present  a  brief  synopsis  of  it.  It  was  first 
observed  in  1772,  again  in  1805,  and  then  a  third  time  in 
1826.  It  was  not  until  this  third  apparition  that  its  peri- 
odicity was  recognized  and  its  previous  appearances  iden- 
tified as  those  of  the  same  body.  The  period  of  revolu- 
tion was  found  to  be  between  six  and  seven  years.  It  was 
so  small  as  to  be  visible  in  ordinary  telescopes  only  when 
the  earth  was  near  it,  which  would  occur  only  at  one  re- 
turn out  of  three  or  four.  So  it  was  not  seen  again  until 
near  the  end  of  1845.  Nothing  remarkable  was  noticed  in 
its  appearance  until  January,  1846,  when-  all  were  aston- 
ished to  find  it  separated  into  two  complete  comets,  one  a 
little  brighter  than  the  other.  The  computation  of  Pro- 
fessor HUBBARD  makes  the  distance  of  the  two  bodies  to 
have  been  200,000  miles. 

The  next  observed  return  was  that  of  1852,  when  the 
two  comets  were  again  viewed,  but  far  more  widely 
separated,  their  distance  having  increased  to  about  a  mil- 
lion and  a  half  of  miles.  Their  brightness  was  so  nearly 
equal  that  it  was  not  possible  to  decide  which  should  be 
considered  the  principal  comet,  nor  to  determine  with 
certainty  which  one  should  be  considered  as  identical  with 
the  comet  seen  during  the  previous  apparition. 

Though  carefully  looked  for  at  every  subsequent  return, 
neither  comet  has  been  seen  since.  In  1872,  Mr.  POGSON, 
of  Madras,  thought  that  he  got  a  momentary  view  of  the 
comet  through  an  opening  between  the  clouds  on  a  stormy 
evening,  but  the  position  in  which  he  supposed  himself  to 
observe  it  was  so  far  from  the  calculated  one  that  his  obser- 
vation has  not  been  accepted. 

Instead  of  the  comet,  however,  we  had  a  meteoric  shower. 
The  orbit  of  this  comet  almost  intersects  that  of  the  earth. 
It  was  therefore  to  be  expected  that  the  latter,  on  passing 


REMARKABLE  COMETS.  405 

the  orbit  of  the  comet,  would  intersect  the  fragmentary 
meteoroids  supposed  to  follow  it,  as  explained  in  the  last 
chapter.  According  to  the  calculated  orbit  of  the  comet,  it 
crossed  the  point  of  intersection  in  September,  1872,  while 
the  earth  passes  the  same  point  on  November  27th  of  each 
year.  It  was  therefore  predicted  that  a  meteoric  shower 
would  be  seen  on  the  night  of  November  27th,  the  radiant 
point  of  which  would  be  in  the  constellation  Andromeda. 
This  prediction  was  completely  verified,  but  the  meteors 
were  so  faint  that  though  they  succeeded  each  other  quite 
rapidly,  they  might  not  have  been  noticed  by  a  casual 
observer.  They  all  radiated  from  the  predicted  point  with 
such  exactness  that  the  eye  could  detect  no  deviation  what- 
ever. 

We  thus  have  a  third  case  in  which  meteoric  showers 
are  associated  with  the  orbit  of  a  comet.  In  this  case,  how- 
ever, the  comet  has  been  completely  dissipated,  and  proba- 
bly has  disappeared  forever  from  telescopic  vision,  though 
it  may  be  expected  that  from  time  to  time  its  invisible 
fragments  will  form  meteors  in  the  earth's  atmosphere. 

§   6.    REMARKABLE   COMETS. 

It  is  familiarly  known  that  bright  comets  were  in  former 
years  objects  of  great  terror,  being  supposed  to  presage 
the  fall  of  empires,  the  death  of  monarchs,  the  approach 
of  earthquakes,  wars,  pestilence,  and  every  other  calamity 
which  could  afflict  mankind.  In  showing  the  entire 
groundlessness  of  such  fears,  science  has  rendered  one  of  its 
greatest  benefits  to  mankind. 

In  1456,  the  comet  known  as  HALLEY'S,  appearing 
when  the  Turks  were  making  war  on  Christendom,  caused 
such  terror  that  Pope  CALIXTUS  ordered  prayers  to  be 
offered  in  the  churches  for  protection  against  it.  This 
is  supposed  to  be  the  origin  of  the  popular  myth  that  the 
Pope  once  issued  a  bull  against  the  comet. 

The  number  of  comets  visible  to  the  naked  eye,  so  far  as 


406 


ASTRONOMY. 


recorded,  has  generally  ranged  from  20  to  40  in  a  cen- 
tury. Only  a  small  portion  #f  these,  however,  have  heeii 
so  bright  as  to  excite  universal  notice. 

Comet  of  1680. — One  of  the  most  remarkable  of  these 
brilliant  comets  is  that  of  1680.  It  inspired  such  terror 
that  a  medal,  of  which  we  present  a  figure,  was  struck 
upon  the  Continent  of  Europe  to  quiet  apprehension.  A 
free  translation  of  the  inscription  is  :  "  The  star  threatens 
evil  things;  trust  only  !  God  will  turn  them  to  good." 
What  makes  this  comet  especially  remarkable  in  history 
is  that  NEWTON  calculated  its  orbit,  and  showed  that  it 
moved  around  the  sun  in  a  conic  section,  in  obedience  to 
the  law  of  gravitation. 


T-RAV. 
GOTT 

WlRDsWoL 

CHEW- 


FlG.    109. — MEDAL   OF   THE    GREAT    COMET   OP   1680. 

Great  Comet  of  1811.  — Fig.  110  shows  its  general  ap- 
pearance. It  has  a  period  of  over  3000  years,  and  its 
aphelion  distance  is  about  40,000,000,000  miles. 

Great  Comet  of  1843. — One  of  the  most  brilliant  com- 
ets which  have  appeared  during  the  present  century  was 
that  of  February,  1843.  It  was  visible  in  full  daylight 
close  to  the  sun.  Considerable  terror  was  caused  in  some 
quarters,  lest  it  should  presage  the  end  of  the  world, 
which  had  been  predicted  for  that  year  by  MILLER.  At 
perihelion  it  passed  nearer  the  sun  than  any  other  body 
has  ever  been  known  to  pass,  the  least  distance  being  only 
about  one  fifth  of  the  sun's  semi-diameter.  With  a  very 
slight  change  of  its  original  motion,  it  would  have  actually 
fallen  into  the  sun. 


GREAT  COMET  OF  1858.  40? 

Great  Comet  of  1858. — Another  remarkable  comet  for 
tlie  length  of  time  it  remained  visible  was  that  of  1858. 
It  is  frequently  called  after  the  name  of  DONATI,  its  first 
discoverer.  No  comet  visiting  our  neighborhood  in 


FlG.    110  — GREAT   COMET   OP    1811. 

recent  times  has  afforded  so  favorable  an  opportunity  for 
studying  its  physical  constitution.  Some  of  the  results  of 
the  observations  made  upon  it  have  already  been  presented. 


111.—  DONATl'S  COMET  OF   1858. 


ENCKE'S  COMET.  409 

Its  greatest  brilliancy  occurred  about  the  beginning  of 
October,  when  its  tail  was  40°  in  length  and  10°  in  breadth 
at  its  outer  end. 

DON  ATI'S  comet  had  not  long  been  observed  when  it 
was  found  that  its  orbit  was  decidedly  elliptical.  After  it 
disappeared,  the  observations  were  all  carefully  investigated 
by  two  mathematicians,  Dr.  VON  ASTEN,  of  Germany, 
and  Mr.  G.  W.  HILL,  of  this  country.  The  latter  found 
a  period  of  1950  years,  which  is  probably  within  a  half  a 
century  of  the  truth.  It  is  probable,  therefore,  that  this 
comet  appeared  about  the  first  century  before  the  Chris- 
tian era,  and  will  return  again  about  the  year  3800. 


Encke's  Comet  and  the  Resisting  Medium. — Of  telescopic 
comets,  that  which  has  been  most  investigated  by  astronomers  is 
known  as  ENCKE'S  comet.  Its  period  is  between  three  and  four 
years.  Viewed  with  a  telescope,  it  is  not  different  in  any  respect 
from  other  telescopic  comets,  appearing  simply  as  a  mass  of  foggy 
light,  somewhat  brighter  near  one  side.  Under  the  most  favorable 
circumstances,  it  is  just  visible  to  the  naked  eye.  The  circumstance 
which  has  lent  most  interest  to  this  comet  is  that  the  observations 
which  have  been  made  upon  it  seem  to  indicate  that  it  is  gradually 
approaching  the  sun.  ENCKE  attributed  this  change  in  its  orbit  to 
the  existence  in  space  of  a  resisting  medium,  so  rare  as  to  have  no 
appreciable  effect  upon  the  motion  of  the  planets,  and  to  be  felt 
only  by  bodies  of  extreme  tenuity,  like  the  telescopic  comets.  The 
approach  of  the  comet  to  the  sun  is  shown,  not  by  direct  obser- 
vation, but  only  by  a  gradual  diminution  of  the  period  of  revolu- 
tion. It  will  be  many  centuries  before  this  period  would  be  so  far 
diminished  that  the  comet  would  actually  touch  the  sun. 

If  the  change  in  the  period  of  this  comet  were  actually  due  to 
the  cause  which  ENCKE  supposed,  then  other  faint  comets  of  the 
same  kind  ought  to  be  subject  to  a  similar  influence.  But  the  in- 
vestigations which  have  been  made  in  recent  times  on  these  bodies 
show  no  deviation  of  the  kind.  It  might,  therefore,  be  concluded 
that  the  change  in  the  period  of  ENCKE 's  comet  must  be  due  to 
some  other  cause.  There  is,  however,  one  circumstance  which 
leaves  us  in  doubt.  ENCKE'S  comet  passes  nearer  the  sun  than  any 
other  comet  of  short  period  which  has  been  observed  with  suffi- 
cient care  to  decide  the  question.  It  may,  therefore,  be  supposed 
that  the  resisting  medium,  whatever  it  may  be,  is  densest  near  the 
sun,  and  does  not  extend  out  far  enough  for  the  other  comets  to 
meet  it.  The  question  is  one  very  difficult  to  settle.  The  fact  is 
that  all  comets  exhibit  slight  anomalies  in  their  motions  which  pre- 
vent us  from  deducing  conclusions  from  them  with  the  same  cer- 
tainty that  we  should  from  those  of  the  planets. 


PART   III. 

THE    UNIVERSE   AT   LARGE. 


INTRODUCTION. 

IN  our  studies  of  the  heavenly  bodies,  we  have  hitherto 
been  occupied  almost  entirely  with  those  of  the  solar  sys- 
tem. Although  this  system  comprises  the  bodies  which 
are  most  important  to  us,  yet  they  form  only  an  insignifi- 
cant part  of  creation.  Besides  the  earth  on  which  we 
dwell,  only  seven  of  the  bodies  of  the  solar  system  are 
plainly  visible  to  the  naked  eye,  whereas  it  is  well  known 
that  2000  stars  or  more  can  be  seen  on  any  clear  night. 
We  now  have  to  describe  the  visible  universe  in  its  largest, 
extent,  and  in  doing  so  shall,  in  imagination,  step  over 
the  bounds  in  which  we  have  hitherto  confined  ourselves 
and  fly  through  the  immensity  of  space. 

The  material  universe,  as  revealed  by  modern  telescopic 
investigation,  consists  principally  of  shining  bodies,  many 
millions  in  number,  a  few  of  the  nearest  and  brightest  of 
which  are  visible  to  the  naked  eye  as  stars.  They  extend 
out  as  far  as  the  most  powerful  telescope  can  penetrate, 
and  no  one  knows  how  much  farther.  Our  sun  is  simply 
one  of  these  stars,  and  does  not,  so  far  as  we  know,  differ 
from  its  fellows  in  any  essential  characteristic.  From  the 
most  careful  estimates,  it  is  rather  less  bright  than  the 
average  of  the  nearer  stars,  and  overpowers  them  by  its 
brilliancy  only  because  it  is  so  much  nearer  to  us. 

The  distance  of  the  stars  from  each  other,  and  therefore 


412  ASTRONOMY. 

from  the  sun,  is  immensely  greater  than  any  of  the  dis- 
tances which  we  have  hitherto  had  to  consider  in  the  solar 
system.  Suppose,  for  instance,  that  a  walker  through 
the  celestial  spaces  could  start  out  from  the  sun,  taking  steps 
3000  miles  long,  or  equal  to  the  distance  from  Liverpool  to 
New  York,  and  making  120  steps  a  minute.  This  speed 
would  carry  him  around  the  earth  in  about  four  seconds  ; 
he  would  walk  from  the  sun  to  the  earth  in  four  hours,  and 
in  five  days  he  would  reach  the  orbit  of  Neptune.  Yet  if 
he  should  start  for  the  nearest  star,  he  would  not  reach  it 
in  a  hundred  years.  Long  before  he  got  there,  the  whole 
orbit  of  Neptune,  supposing  it  a  visible  object,  would 
have  been  reduced  to  a  point,  and  finally  vanish  from 
sight  altogether.  In  fact,^the  nearest  known  star  is  about 
seven  thousand  times  as  far  as  the  planet  Neptune.  If 
we  suppose  the  orbit  of  this  planet  to  be  represented  by  a 
child's  hoop,  the  nearest  star  would  be  three  or  four  miles 
away.  We  have  no  reason  to  suppose  that  contiguous 
stars  are,  on  the  average,  nearer  than  this,  except  in  special 
cases  where  they  are  collected  together  in  clusters. 

The  total  number  of  the  stars  is  estimated  by  millions, 
and  they  are  probably  separated  by  these  wide  intervals. 
It  follows  that,  in  going  from  the  sun  to  the  nearest  star, 
we  would  be  simply  taking  one  step  in  the  universe.  The 
most  distant  stars  visible  in  great  telescopes  are  probably 
several  thousand  times  more  distant  than  the  nearest  one, 
and  we  do  not  know  what  may  lie  beyond. 

The  point  we  wish  principally  to  impress  on  the  reader 
in  this  connection  is  that,  although  the  stars  and  planets  pre- 
sent to  the  naked  eye  so  great  a  similarity  in  appearance, 
there  is  the  greatest  possible  diversity  in  their  distances 
and  characters.  The  planets,  though  many  millions  of 
miles  away,  are  comparatively  near  us,  and  form  a  little 
family  by  themselves,  which  is  called  the  solar  system. 
The  fixed  stars  are  at  distances  incomparably  greater — the 
nearest  star,  as  just  stated,  being  thousands  of  times  more 
distant  than  the  farthest  planet.  The  planets  are,  so  far 


THE  UNIVERSE  AT  LARGE.  413 

as  we  can  see,  worlds  somewhat  like  this  on  which  we  live, 
while  the  stars  are  suns,  generally  larger  and  brighter  than 
our  own.  Each  star  may,  for  aught  we  know,  have  plan- 
ets revolving  around  it,  but  their  distance  is  so  immense 
that  the  largest  planets  will  remain  invisible  with  the  most 
powerful  telescopes  man  can  ever  hope  to  construct. 

The  classification  of  the  heavenly  bodies  thus  leads  us  to 
this  curious  conclusion.  Our  sun  is  one  of  the  family  of 
stars,  the  other  members  of  which  stud  the  heavens  at 
night,  or,  in  other  words,  the  stars  are  suns  like  that  which 
makes  the  day.  The  planets,  though  they  look  like  stars, 
are  not  such,  but  bodies  more  like  the  earth  on  which 
we  live. 

The  great  universe  of  stars,  including  the  creation  in  its 
largest  extent,  is  called  the  stellar  system,  or  stellar 
universe.  We  have  first  to  consider  how  it  looks  to  the 
naked  eye. 


CHAPTER  I. 

THE   CONSTELLATIONS. 
§   1.    GENERAL  ASPECT  OP  THE  HEAVENS. 

WHEN  we  view  the  heavens  with  the  unassisted  eye,  the 
stars  appear  to  be  scattered  nearly  at  random  over  the 
surface  of  the  celestial  vault.  The  only  deviation  from  an 
entirely  random  distribution  which  can  be  noticed  is  a  cer- 
tain grouping  of  the  brighter  ones  into  constellations. 
We  notice  also  that  a  few  are  comparatively  much  brighter 
than  the  rest,  and  that  there  is  every  gradation  of  bril- 
liancy, from  that  of  the  brightest  to  those  which  are  barely 
visible.  We  also  notice  at  a  glance  that  the  fainter  stars 
outnumber  the  bright  ones  ;  so  that  if  we  divide  the  stars 
into  classes  according  to  their  brilliancy,  the  fainter  classes 
will  be  far  the  more  numerous. 

The  total  number  one  can  see  will  depend  very  largely 
upon  the  clearness  of  the  atmosphere  and  the  keenness  of 
the  eye.  From  the  most  careful  estimates  which  have 
been  made,  it  would  appear  that  there  are  in  the  whole 
celestial  sphere  about  6000  stars  visible  to  an  ordinarily 
good  eye.  Of  these,  however,  we  can  never  see  more  than 
a  fraction  at  any  one  time,  because  one  half  of  the  sphere  is 
always  of  necessity  below  the  horizon.  If  we  could  see  a 
star  in  the  horizon  as  well  as  in  the  zenith,  one  half  of  the 
whole  number,  or  3000,  would  be  visible  on  any  clear  night. 
But  stars  near  the  horizon  are  seen  through  so  great  a 
thickness  of  atmosphere  as  greatly  to  obscure  their  light  ; 
consequently  only  the  brightest  ones  can  there  be  seen.  As 


CLASSES  OF  STARS.  415 

a  result  of  this  obscuration,  it  is  not  likely  that  more  than 
2000  stars  can  ever  be  taken  in  at  a  single  view  by  any 
ordinary  eye.  About  2000  other  stars  are  so  near  the 
South  Pole  that  they  never  rise  in  our  latitudes.  Hence 
out  ol  the  6000  supposed  to  be  visible,  only  4000  ever 
come  within  the  range  of  our  vision,  unless  we  make  a 
journey  toward  the  equator. 

The  Galaxy. — Another  feature  of  the  heavens,  which  is 
less  striking  than  the  stars,  but  has  been  noticed  from 
the  earliest  times,  is  the  Galaxy,  or  Milky  Way.  This 
object  consists  of  a  magnificent  stream  or  belt  of  white 
milky  light  10°  or  15°  in  breadth,  extending  obliquely 
around  the  celestial  sphere.  During  the  spring  months,  it 
nearly  coincides  with  our  horizon  in  the  early  evening, 
but  it  can  readily  be  seen  at  all  other  times  of  the  year 
spanning  the  heavens  like  an  arch.  It  is  for  a  portion  of 
its  length  split  longitudinally  into  two  parts,  which  remain 
separate  through  many  degrees,  and  are  finally  united 
again.  The  student  will  obtain  a  better  idea  of  it  by 
actual  examination  than  from  any  description.  He  will 
see  that  its  irregularities  of  form  and  lustre  are  such  that 
in  some  places  it  looks  like  a  mass  of  brilliant  clouds.  In 
the  southern  hemisphere  there  are  vacant  spaces  in  it 
which  the  navigators  call  coal-sacks.  In  one  of  these, 
5°  by  18°,  there  is  scarcely  a  single  star  visible  to  the 
naked  eye  (see  Figs.  121  and  132). 

Lucid  and  Telescopic  Stars.  —  When  we  view  the 
heavens  with  a  telescope,  we  find  that  there  are  innumer- 
able stars  too  small  to  be  seen  by  the  naked  eye.  We 
may  therefore  divide  the  stars,  with  respect  to  brightness, 
into  two  great  classes. 

Lucid  Stars  are  those  which  are  visible  without  a  tele- 
scope. 

Telescopic  Stars  are  those  which  are  not  so  visible. 

When  GALILEO  first  directed  his  telescope  to  the  heav- 
ens, about  the  year  1610,  he  perceived  that  the  Milky 
Way  was  composed  of  stars  too  faint  to  be  individually 


416  ASTRONOMY. 

seen  by  the  unaided  eye.  We  thus  have  the  interesting 
fact  that  although  telescopic  gtars  cannot  be  seen  one  by 
one,  yet  in  the  region  of  the  Milky  Way  they  are  so  numer- 
ous that  they  shine  in  masses  like  brilliant  clouds.  HUY- 
GHENS  in  1 656  resolved  a  large  portion  of  the  Galaxy  into 
stars,  and  concluded  that  it  was  composed  entirely  of  them. 
KEPLER  considered  it  to  be  a  vast  ring  of  stars  surround- 
ing the  solar  system,  and  remarked  that  the  sun  must  be 
situated  near  the  centre  of  the  ring.  This  view  agrees 
very  well  with  the  one  now  received,  only  that  the  stars 
which  form  the  Milky  Way,  instead  of  lying  around  the 
solar  system,  are  at  a  distance  so  vast  as  to  elude  all  our 
powers  of  calculation. 

Such  are  in  brief  the  more  salient  phenomena  which 
are  presented  to  an  observer  of  the  starry  heavens.  We 
shall  now  consider  how  these  phenomena  have  been  clas- 
sified by  an  arrangement  of  the  stars  according  to  their 
brilliancy  and  their  situation. 

§   2.    MAGNITUDES  OP    THE    STABS. 

In  ancient  times,  the  stars  were  arbitrarily  classified  into 
six  orders  of  magnitude.  The  fourteen  brightest  visible  in 
our  latitude  were  designated  as  of  the  first  magnitude,  while 
those  which  were  barely  visible  to  the  naked  eye  were  said 
to  be  of  the  sixth  magnitude.  This  classification,  it  will 
be  noticed,  is  entirely  arbitrary,  since  there  are  no  two 
stars  which  are  absolutely  of  the  same  brilliancy,  while  if 
all  the  stars  were  arranged  in  the  order  of  their  actual 
brilliancy,  we  should  find  a  regular  gradation  from  the 
brightest  to  the  faintest,  no  two  being  precisely  the  same. 
Therefore  the  brightest  star  of  any  one  magnitude  is 
about  of  the  same  brilliancy  with  the  faintest  one  of  the 
next  higher  magnitude.  It  depends  upon  the  judgment 
of  the  observer  to  what  magnitude  a  given  star  shall  be 
assigned  ;  so  that  we  cannot  expect  an  agreement  on  this 
point.  The  most  recent  and  careful  division  into  niagni- 


MAGNITUDES  OF  STARS.  417 

tildes  has  been  made  by  HEIS,  of  Germany,  whose  results 
with  respect  to  numbers  are  as  follows.  Between  the 
North  Pole  and  35°  south  declination,  there  are  : 

14  stars  of  the  first  magnitude, 

48  "  "  second  " 
152  "  "  third  " 
313  "  "  fourth  " 

854     "       "        fifth 
3974     "       "        sixth         " 


5355  of  the  first  six  magnitudes. 

Of  these,  however,  nearly  2000  of  the  sixth  magnitude 
are  so  faint  that  they  can  be  seen  only  by  an  eye  of  extra- 
ordinary keenness. 

In  order  to  secure  a  more  accurate  classification  and  expression  of 
brightness,  HEIS  and  others  have  divided  each  magnitude  into 
three  orders  or  sub-magnitudes,  making  eighteen  orders  in  all 
visible  to  the  naked  eye.  When  a  star  was  considered  as  falling  be- 
tween two  magnitudes,  both  figures  were  written,  putting  the  mag- 
nitude to  which  the  star  most  nearly  approached  first.  For  in- 
stance, the  faintest  stars  of  the  fourth  magnitude  were  called  4 '5. 
The  next  order  below  this  would  be  the  brightest  of  the  fifth 
magnitude  ;  these  were  called  5 '4.  The  stars  of  the  average  fifth 
magnitude  were  called  5  simply.  The  fainter  ones  were  called  5-6, 
and  so  on.  This  notation  is  still  used  by  some  astronomers,  but 
those  who  aim  at  greater  order  and  precision  express  the  magni- 
tudes in  tenths.  For  instance,  the  faintest  stars  of  the  fifth  magni- 
tude they  would  call  4  •  6,  those  one  tenth  fainter  4  •  7,  and  so  on 
until  they  reached  the  average  of  the  fifth  magnitude,  which 
would  be*  5-0.  The  division  into  tenths  of  magnitudes  is  as  mi- 
nute a  one  as  the  ordinary  eye  is  able  to  make. 

This  method  of  designating  the  brilliancy  of  a  star  on  a  scale  of 
magnitudes  is  not  at  all  accurate.  Several  attempts  have  been 
made  in  recent  times  to  obtain  more  accurate  determinations,  by 
measuring  the  light  of  the  stars.  A.n  instrument  with  which  this 
can  be  done  is  called  a  photometer.  The  results  obtained  with  the 
photometer  have  been  used  to  correct  the  scale  of  magnitudes 
and  make  it  give  a  more  accurate  expression  for  the  light  of  the 
stars.  The  study  of  such  measures  shows  that,  for  the  most  part, 
the  brightness  of  the  stars  increases  in  geometrical  progression  as 
the  magnitudes  vary  in  arithmetical  progression.  The  stars  of  one 
magnitude  are  generally  about  2£  times  as  bright  as  those  of  the 
magnitude  next  below  it.  Therefore  if  we  take  the  light  of  a  star 


418  ASTRONOMY. 

of  the  sixth  magnitude,  wThich  is  just  visible  to  the  naked  eye,  as 
unity,  we  shall  have  the   following  scale  : 

Magnitude  6th,  brightness     1 
5th, 


u 


4th, 
3d, 
2d, 
1st, 


6* 

16  nearly 
40 
100 


Therefore,  according  to  these  estimates,  an  average  star  of  the 
first  magnitude  is  about  100  times  as  bright  as  one  of  the  sixth. 
There  is,  however,  a  deviation  from  this  scale  in  the  case  of  the 
brighter  magnitudes,  an  average  star  of  the  second  magnitude 
being  perhaps  three  times  as  bright  as  one  of  the  third,  and  most 
of  the  stars  of  the  first  magnitude  brighter  than  those  of  the  second 
in  a  yet  larger  ratio.  Indeed,  the  first  magnitude  stars  differ  so 
greatly  in  brightness  that  we  cannot  say  how  bright  a  standard 
star  of  that  magnitude  really  is.  Sirius,  for  instance,  is  probably 
500  times  as  bright  as  a  sixth  magnitude  star. 

The  logarithm  of  2£  being  very  nearly  0'40,  we  can  readily  find 
how  many  stars  of  any  one  magnitude  are  necessary  to  make  one  of 
the  higher  magnitude  by  multiplying  the  difference  of  the  magni- 
tude by  0*40,  and  taking  the  number  corresponding  to  this  logarithm. 

This  scale  will  enable  us  to  calculate  in  a  rough  way  the  magni- 
tude of  the  smallest  stars  which  can  be  seen  with  a  telescope  of  given 
aperture.  The  quantity  of  light  which  a  telescope  admits  is  directly 
as  the  square  of  its  aperture.  The  amount  of  light  emitted  by  the 
faintest  star  visible  in  it  is  therefore  inversely  as  this  square.  If  we 
increase  the  aperture  50  per  cent,  we  increase  the  seeing  power  of 
our  telescope  about  one  magnitude.  More  exactly,  the  ratio  of  in- 
crease of  aperture  is  V  2|,  or  1  •  58.  The  pupil  of  the  eye  is  probably 
equivalent  to  a  telescope  of  about  ^  of  an  inch  in  aperture  ;  that 
is,  in  a  telescope  of  this  size  the  faintest  visible  star  would  be  about 
of  the  sixth  magnitude.  To  find  the  exact  magnitude  of  the 
faintest  star  visible  with  a  larger  telescope,  we  recall  that  the 
quantity  of  light  received  by  the  objective  is  proportional  to  the 
square  of  the  aperture.  As  just  shown,  every  time  we  multiply  the 
square  of  the  aperture  by  2|,  or  the  aperture  itself  by  the  square 
root  of  this  quantity,  we  add  one  magnitude  to  the  power  of  our 
telescope.  Therefore,  if  we  call  a0  the  aperture  of  a  telescope 
which  would  just  show  a  star  one  magnitude  brighter  than  the 
first  (or  mag.  0),  the  aperture  necessary  to  show  a  star  of  magnitude 
m  will  be  found  by  mltiplying  «„  by  1-58  m  times— that  is,  it  will 
be  1.58m  a0.  So,  calling  a  this  aperture,  we  have  : 

a  =  l- 58m  ffi0  =  a0  V  2-5™. 

Taking  the  logarithms  of  both  sides  of  the  equation,  and  using  ap- 
proximate round  numbers  which  are  exact  enough  for  this  purpose  : 

log.  a  =  m  log.  1  -58  +  log.  a0  =  ^  log.  2-5  +  log.  a0  =  ^  +  log.  a0. 


NAMES  OF  THE  STARS. 


419 


Now,   as  just  found,  when  m  =  6,   a  =  Oin-25  =  6-4  millimetres. 
With  these  values  of  a  and  m  we  find  : 

log.  «0  =  —  1-802  in  fractions  of  an  inch. 

=  —  0-397  in  fractions  of  a  millimetre. 

Hence,  when  the  magnitude  is  given,  and  we  wish  to  find  the  aperture  : 
log.  a  =  —   —  1-802  [will  give  aperture  in  inches.] 

log.  a  =  —   —  0-897  [will  give  aperture  in  millimetres. j 
5 

If  the  aperture  is  given,  and  we  require  the  limiting  magnitude  . 

m  =  5  log.  a  4-  9-0  [if  a  is  in  inches.] 

m  =  5  log.  a  +  2-0  [if  a  is  in  millimetres.] 

The  magnitudes  for  different  apertures  is  shown  in  the  following 
table : 


Aperture. 

Minimum 
Visibile. 

Aperture. 

Minimum 
Visibile. 

Inches. 

Magnitude. 

Inches. 

Magnitude. 

1-0 

9-0 

6-5 

13-1 

1-5 

9-9 

7-0 

13-3 

2-0 

10-5 

8-0 

13-5 

2-5 

11-0 

9-0 

13-8 

3-0 

11-4 

10-0 

14-0 

3-5 

11-7 

11-0 

14-2 

4-0 

12-0 

12-0 

14-4 

4-5 

12-3 

15-0 

14-9 

5-0 

12-5 

18-0 

15-3 

55 

12-7 

26-0 

16-1 

0-0 

12-9 

34-0 

16-6 

§  3.  THE  CONSTELLATIONS  AND  NAMES  OP  THE 

STABS. 

The  earliest  astronomers  divided  the  stars  into  groups, 
called  constellations,  and  gave  special  proper  names  both 
to  these  groups  and  to  many  of  the.  more  conspicuous 
stars.  We  have  no  record  of  the  process  by  which  this 
was  done,  or  of  the  considerations  which  led  to  it.  It  was 
long  before  the  commencement  of  history,  as  we  may  in- 
fer from  different  allusions  to  the  stars  and  constellations 
in  the  book  of  Job,  which  is  supposed  to  be  among  the 


420  ASTRONOMY. 

most  ancient  writings  now  extant.  We  have  evidence 
that  more  than  3000  years  before  the  commencement  of 
the  Christian  chronology  the  star  Sirius,  the  brightest  in 
the  heavens,  was  known  to  the  Egyptians  under  the  name 
of  Sothis.  Arcturus  is  mentioned  by  JOB  himself.  The 
seven  stars  of  the  Great  Bear,  so  conspicuous  in  our  north- 
ern sky,  were  known  under  that  name  to  HOMER  and  HE- 
SIOD,  as  well  as  the  group  of  the  Pleiades,  or  Seven  Stars, 
and  the  constellation  of  Orion.  Indeed,  it  would  seem 
that  all  the  earlier  civilized  nations,  Egyptians,  Chinese, 
Greeks,  and  Hindoos,  had  some  arbitrary  division  of  the 
surface  of  the  heavens  into  irregular,  and  often  fantastic 
shapes,  which  were  distinguished  by  names. 

In  early  times,  the  names  of  heroes  and  animals  were 
given  to  the  constellations,  and  these  designations  have 
come  down  to  the  present  day.  Each  object  was  sup- 
posed to  be  painted  on  the  surface  of  the  heavens,  and  the 
stars  were  designated  by  their  position  upon  some  portion 
of  the  object.  The  ancient  and  mediaeval  astronomers 
would  speak  of  "  the  bright  star  in  the  left  foot  of 
Orion, "  "  the  eye  of  the  Bull, "  "  the  heart  of  the  Lion, '  > 
"  the  head  of  Perseus"  etc.  These  figures  are  still  re- 
tained upon  some  star-charts,  and  are  useful  where  it  is 
desired  to  compare  the  older  descriptions  of  the  constella- 
tions with  our  modern  maps.  Otherwise  they  have  ceased 
to  serve  any  purpose,  and  are  not  generally  found  on  maps 
designed  for  astronomical  uses. 

The  Arabians,  who  used  this  clumsy  way  of  designating 
stars,  gave  special  names  to  a  large  number  of  the  brighter 
ones.  Some  of  these  names  are  in  common  use  at  the 
present  time,  as  Aldebaran,  Fomalliaut,  etc.  A  few  other 
names  of  bright  stars  have  come  down  from  prehistoric 
times,  that  of  Arcturus  for  instance  :  they  are,  how- 
ever, gradually  falling  out  of  use,  a  system  of  nomencla- 
ture introduced  in  modern  times  having  been  substituted. 

In  1654,  BAYER,  of  Germany,  mapped  down  the  constel- 
lations upon  charts,  designating  the  brighter  stars  of  each 


NAMING  THE  STARS.  421 

constellation  by  the  letters  of  the  Greek  alphabet.  When 
this  alphabet  was  exhausted,  he  introduced  the  letters  of 
the  Roman  alphabet.  In  general,  the  brightest  star  was 
designated  by  the  first  letter  of  the  alphabet  a,  the  next 
by  the  following  letter  (3,  etc.  Although  this  is  sometimes 
supposed  to  have  been  his  rule,  the  Greek  letter  affords 
only  an  imperfect  clue  to  the  average  magnitude  of  a  star. 
In  a  great  many  of  the  constellations  there  are  deviations 
from  the  order,  the  brightest  star  being  (3 ;  but  where  stars 
differ  by  an  entire  magnitude  or  more,  the  fainter  ones 
nearly  always  follow  the  brighter  ones  in  alphabetical  order. 

On  this  system,  a  star  is  designated  by  a  certain  Greek 
letter,  followed  by  the  genitive  of  the  Latin  name  of  the 
constellation  to  which  it  belongs.  For  example,  a  Canis 
Majoris,  or,  in  English,  a.  of  the  Great  Dog,  is  the  desig- 
nation of  /Sirius,  the  brightest  star  in  the  heavens.  The 
seven  stars  of  the  Great  Bear  are  called  OL  Ursce  Jfajoris, 
(3  Ursce  Majoris,  etc.  Arcturus  is  a  Bootis.  The 
reader  will  here  see  a  resemblance  to  our  way  of  designat- 
ing individuals  by  a  Christian  name  followed  by  the  family 
name.  The  Greek  letters  furnish  the  Christian  names  of 
the  separate  stars,  while  the  name  of  the  constellation  is 
that  of  the  family.  As  there  are  only  fifty  letters  in  the 
two  alphabets  used  by  BAYER,  it  will  be  seen  that  only  the 
fifty  brightest  stars  in  each  constellation  could  be  desig- 
nated by  this  method.  In  most  of  the  constellations  the 
number  thus  chosen  is  much  less  than  fifty. 

When  by  the  aid  of  the  telescope  many  more  stars  than 
these  were  laid  down,  some  other  method  of  denoting 
them  became  necessary.  FLAMSTEED,  who  observed  be- 
fore and  after  1700,  prepared  an  extensive  catalogue  of 
stars,  in  which  those  of  each  constellation  were  designated 
by  numbers  in  the  order  of  right  ascension.  These  num- 
bers were  entirely  independent  of  the  designations  of 
BAYER — that  is,  he  did  not  omit  the  BAYER  stars  from 
his  system  of  numbers,  but  numbered  them  as  if  they  had 
no  Greek  letter.  Hence  those  stars  to  which  BAYER  ap- 


422  ASTRONOMY. 

plied  letters  have  two  designations,   the  letter  and  the 
number. 

FLAMSTEED'S  numbers  do  not  go  much  above  100  for 
any  one  constellation — Taurus,  the  richest,  having  139. 
When  we  consider  the  more  numerous  minute  stars,  no 
systematic  method  of  naming  them  is  possible.  The  star 
can  be  designated  only  by  its  position  in  the  heavens,  or 
the  number  which  it  bears  in  some  well-known  catalogue. 

§  4.    DESCRIPTION  OF  THE  CONSTELLATIONS. 

The  aspect  of  the  starry  heavens  is  so  pleasing  that 
nearly  every  intelligent  person  desires  to  possess  some 
knowledge  of  the  names  and  forms  of  the  principal  con- 
stellations. We  therefore  present  a  brief  description  of 
the  more  striking  ones,  illustrated  by  figures,  so  that  the 
reader  may  be  able  to  recognize  them  when  he  sees  them 
on  a  clear  night. 

We  begin  with  the  constellations  near  the  pole,  because 
they  can  be  seen  on  any  clear  night,  while  the  southern 
ones  can,  for  the  most  part,  only  be  seen  during  certain 
seasons,  or  at  certain  hours  of  the  night.  The  accompany- 
ing figure  shows  all  the  stars  within  50°  of  the  pole  down 
to  the  fourth  magnitude  inclusive.  The  Roman  numerals 
around  the  margin  show  the  meridians  of  right  ascension, 
one  for  every  hour.  In  order  to  have  the  map  represent 
the  northern  constellations  exactly  as  they  are,  it  must  be 
held  so  that  the  hour  of  sidereal  time  at  which  the  observer 
is  looking  at  the  heavens  shall  be  at  the  top  of  the  map. 
Supposing  the  observer  to  look  at  nine  o'clock  in  the  even- 
ing, the  months  around  the  margin  of  the  map  show  the 
regions  near  the  zenith.  He  has  therefore  only  to  hold  the 
map  with  the  month  upward  and  face  the  north,  when  he 
will  have  the  northern  heavens  as  they  appear,  except 
that  the  stars  near  the  bottom  of  the  map  will  be  cut  off 
by  the  horizon. 

The  first  constellation  to  be  looked  for  is  Ursa  Major, 


THE   CONSTELLATIONS. 


433 


the  Great  Beour,  familiarly  known  as  "  the  Dipper."  Ihe 
two  extreme  stars  in  this  constellation  point  toward  the 
pole-star,  as  already  explained  in  the  opening  chapter. 

Ursa  Minor  y  sometimes  called  ' '  the  Little  Dipper, ' '  is 
the  constellation  to  which  the  pole-star  belongs.     About 


FlG.    112. — MAP  OP  THE  NORTHERN  CONSTELLATIONS. 

15°  from  the  pole,  in  right  ascension  XV.  hours,  is  a  star 
of  the  second  magnitude,  ft  Ursce  Minoris,  about  as  bright 
as  the  pole-star.  A  curved  row  of  three  small  stars  lies 
between  these  two  bright  ones,  and  forms  the  handle  of 
the  supposed  dipper. 


424  ASTRONOMT. 

Cassiopeia,  or  "  the  Lady  in  the  Chair,"  is  near  hour  I 
of  right  ascension,  on  the  opposite  side  of  the  pole-star 
from  Ursa  Major,  and  at  nearly  the  same  distance. 
The  six  brighter  stars  are  supposed  to  bear  a  rude  resem- 
blance to  a  chair.  In  mythology,  Cassiopeia  was  the  queen 
of  Cepheus,  and  in  the  mythological  representation  of  the 
constellation  she  is  seated  in  the  chair  from  which  she  is 
issuing  her  edicts. 

In  hour  III  of  right  ascension  is  situated  the  constella- 
tion Perseus,  about  10°  further  from  the  pole  than  Cas- 
siopeia. The  Milky  Way  passes  through  these  two  con- 
stellations. 

Draco,  the  Dragon,  is  formed  principally  of  a  long 
row  of  stars  lying  between  Ursa  Major  and  Ursa  Minor. 
The  head  of  the  monster  is  formed  of  the  northernmost 
three  of  four  bright  stars  arranged  at  the  corners  of  a 
lozenge  between  XVII  and  XYIII  hours  of  right  ascen- 
sion. 

Cepheus  is  on  the  opposite  side  of  Cassiopeia  from 
Perseus,  lying  in  the  Milky  Way,  about  XXII  hours  of 
right  ascension.  It  is  not  a  brilliant  constellation. 

Other  constellations  near  the  pole  are  Camelopardalis, 
Lynx,  and  Lacerta  (the  Lizard),  but  they  contain  only 
small  stars. 

In  describing  the  southern  constellations,  we  shall  take 
four  separate  positions  of  the  starry  sphere  corresponding 
respectively  to  YI  hours,  XII  hours,  XYIII  hours, 
and  0  hours  of  sidereal  time  or  right  ascension.  These 
hours  of  course  occur  every  day,  but  not  always  at  con- 
venient times,  because  they  vary  with  the  time  of  the 
year,  as  explained  in  Chapter  I. ,  Part  I. 

We  shall  first  suppose  the  observer  to  view  the  heavens 
at  YI  hours  of  sidereal  time,  which  occurs  on  Decem- 
ber 21st  about  midnight,  January  1st  about  11.30  P.M., 
February  1st  about  9.30  P.M.,  March  1st  about  7.30 
P.M.,  and  so  on  through  the  year,  two  hours  earlier  every 
month.  In  this  position  of  the  sphere,  the  Milky  Way 


THE  CONSTELLATIONS.  425 

spans  the  heavens  like  an  arch,  resting  on  the  horizon  be- 
tween north  and  north-west  on  one  side,  and  between 
south  and  south-east  on  the  other.  We  shall  first  describe 
the  constellations  which  lie  in  its  course,  beginning  at  the 
north.  Cepheus  is  near  the  north-west  horizon,  and  above 
it  is  Cassiopeia,  distinctly  visible  at  an  altitude  nearly 
equal  to  that  of  the  pole.  Next  is  Perseus,  just  north- 
west of  the  zenith.  Above  Perseus  lies  Auriga,  the 
Charioteer,  which  may  be  recognized  by  a  bright  star  of 
the  first  magnitude  called  Capella  (the  Goat),  now  quite 
near  the  zenith.  Auriga  is  represented  as  holding  a 
goat  in  his  arms,  in  the  body  of  which  the  star  is  situated. 
About  10°  east  of  Capella  is  the  star  (3  Aurigw  of  the 
second  magnitude. 

Going  further  south,  the  Milky  Way  next  passes  between 
Taurus  and  Gemini. 

Taurus,  the  Bull,  may  be  recognized  by  the  Pleiades, 
or  "  Seven  Stars."  Really  there  are  only  six  stars  in  the 
group  clearly  visible  to  ordi- 
nary eyes,  and  any  eye  strong 
enough  to  see  seven  will  prob- 
ably see  four  others,  or  eleven 
in  all.  This  group  forms  an 
interesting  object  of  study 
with  a  small  telescope,  as  sixty 
or  eighty  stars  can  then  readily 
be  seen.  We  therefore  pre- 
sent a  telescopic  view  of  it, 
the  six  large  stars  being  those 
visible  to  any  ordinary  eye, 
the  five  next  in  size  those 
which  can  be  seen  by  a  re-  Fl<>-  113. —TELESCOPIC  VIEW  OP 

T      ,  T  ,  ,     ,-.  THE   PLEIADES. 

markably  good  eye,  and  the 

others  those  which  require  a  telescope.  East  of  the  Pleia- 
des is  the  bright  red  star  Aldebaran,  or  "  the  Eye  of 
the  Bull."  It  lies  in  a  group  called  the  Ilyades,  ar- 
ranged in  the  form  of  the  letter  V,  and  forming  the  face 


426 


ASTRONOMY. 


of  the  Bull.  In  the  middle  of  one  of  the  legs  of  the  V 
will  be  seen  a  beautiful  pair  of  stars  of  the  fourth  magni- 
tude very  close  together.  Tfiey  are  called  6  Tauri. 

Gemini,  the  Twins,  lie  east  of  the  Milky  Way,  and 
may  be  recognized  by  the  bright  stars  Castor  and  Pollux, 
which  lie  20°  or  30°  south-east  or  south  of  the  zenith. 


FlG.    114. — THE  CONSTELLATION    ORION. 

They  are  about  5°  apart,   and  Pollux,  the  southernmost 
one,  is  a  little  brighter  than  Castor. 

Orion,  the  most  brilliant  constellation  in  the  heavens, 
is  very  near  the  meridian,  lying  south-east  of  Taurus  and 
south-west  of  Gemini.  It  may  be  readily  recognized  by 
the  figure  which  we  give.  Four  of  its  bright  stars  form 


THE  CONSTELLATIONS.  42? 

the  corners  of  a  rectangle  about  15°  long  from  north 
to  south,  and  5°  wide.  In  the  middle  of  it  is  a  row  of 
three  bright  stars  of  the  second  magnitude,  which  no  one 
can  fail  to  recognize.  Below  this  is  another  row  of  three 
smaller  ones.  The  middle  star  of  this  last  row  is  called 
0  Orionis,  and  is  situated  in  the  midst  of  the  great  nebula 
of  Orion,  one  of  the  most  remarkable  telescopic  objects  in 
the  heavens.  Indeed,  to  the  naked  eye  this  star  has  a 
nebulous  hazy  appearance.  The  two  stars  of  the  first 
magnitude  are  a  Orionis,  or  Betelguese,  which  is  the  high- 
est, and  may  be  recognized  by  its  red  color,  and  Rigel, 
or  ft  Orionis,  a  sparkling  white  star  lower  down  and  a 
little  to  the  west.  The  former  is  in  the  shoulder  of  the 
figure,  the  latter  in  the  foot.  A  little  north-west  of 
Betelguese  are  three  small  stars,  which  form  the  head. 
The  row  of  stars  on  the  west  form  his  arm  and  club,  the 
latter  being  raised  as  if  to  strike  at  Taurus,  the  Bull,  on 
the  west. 

Canis  Minor,  the  Little  Bog,  lies  across  the  Milky 
Way  from  Orion,  and  may  be  recognized  by  the  bright 
star  Procyon  of  the  first  magnitude.  The  three  stars 
Pollux,  Procyon,  and  Betelguese  form  a  right-angled  tri- 
angle, the  right  angle  being  at  Procyon. 

Canis  Major,  the  Great  Dog,  lies  south-east  of  Orion, 
and  is  easily  recognized  by  Sirius,  the  brightest  fixed  star 
in  the  heavens.  A  number  of  bright  stars  south  and 
south-east  of  Sirius  belong  to  this  constellation,  making 
it  one  of  great  brilliancy. 

Argo  Navis,  the  ship  Argo,  lies  near  the  south  horizon, 
partly  above  it  and  partly  below  it.  Its  brightest  star  is 
Canopus,  which,  next  to  Sirius,  is  the  brightest  star  in 
the  heavens.  Being  in  53°  of  south  decimation,  it  never 
rises  to  an  observer  within  53°  of  the  North  Pole — that  is, 
north  of  37°  of  north  latitude.  In  our  country  it  is  visi- 
ble only  in  the  Southern  States,  and  even  there  only 
between  six  and  seven  hours  of  sidereal  time, 

We  next  trace  out  the  zodiacal  constellations,  which  are 


428  ASTRONOMY. 

of  interest  because  it  is  through  them  that  the  sun  passes 
in  its  apparent  annual  course.  We  shall  commence  in 
the  west  and  go  toward  the  east,  in  the  order  of  right 
ascension. 

Aries,  the  Ram,  is  in  the  west,  about  one  third  of  the 
way  from  the  horizon  to  the  zenith.  It  may  be  recognized 
by  three  stars  of  the  second,  third,  and  fourth  magni- 
tudes, respectively,  forming  an  obtuse-angled  triangle. 
The  brightest  star  is  the  highest.  Next  toward  the  east 
is  Taurus,  the  Bull,  which  brings  us  nearly  to  the  meri- 
dian, and  east  of  the  meridian  lies  Gemini,  the  Twins,  both 
of  which  constellations  have  just  been  described. 


FlG.    115. — THE   CONSTELLATION   LEO,    THE  LION. 

Cancer,  the  Crab,  lies  east  of  Gemini,  but  contains  no 
bright  star.  The  most  noteworthy  object  in  this  constel- 
lation is  Prcesepe,  a  group  of  telescopic  stars,  which  ap- 
pears to  the  naked  eye  like  a  spot  of  milky  light.  To  see 
it  well,  the  night  must  be  clear  and  the  moon  not  in  the 
neighborhood. 

Leo,  the  Lion,  is  from  one  to  two  hours  above  the 
eastern  horizon.  Its  brightest  star  is  Eegulus,  one  third 
of  the  way  from  the  eastern  horizon  to  the  zenith,  and 
between  the  first  and  second  magnitudes.  Five  or  six 
stars  north  of  it  in  a  curved  line  are  in  the  form  of  a 


THE  CONSTELLA  T10NS.  429 

sickle,  of  which  Regulus  is  the  handle.  As  the  Lion  was 
drawn  among  the  old  constellations,  Itegulus  formed  his 
heart,  and  was  therefore  called  Cor  Leonis.  The  sickle 
forms  his  head,  and  his  body  and  tail  extend  toward  the 
horizon.  The  tail  ends  near  the  star  Denebola,  which  is 
quite  near  the  horizon. 

Leo  Minor  lies  to  the  north  of  Leo,  and  Sextans,  the 
Sextant,  south  of  it,  but  neither  contains  any  bright  stars. 

Eridanus,  the  River  Po,  south-west  of  Orion  ;  Lepus, 
the  Hare,  south  of  Orion  and  west  of  Canis  Major  • 
Columba,  the  Dove,  south  of  Lepus,  are  constellations  in 
the  south  and  south-west,  which,  however,  have  no  strik- 
ing features. 

The  constellations  we  have  described  are  those  seen  at 
six  hours  of  sidereal  time.  If  the  sky  is  observed  at  some 
other  hour  near  this,  we  may  find  the  sidereal  time  by  the 
rule  given  in  Chapter  I.,  §  5,  p.  30,  and  allow  for  the  di- 
urnal motion  during  the  interval. 

Appearance  of  the  Constellations,  at  12  Hours  Sidereal 
Time. — This  hour  occurs  on  April  1st  at  11.30  P.M.,  on 
May  1st  at  9.30  P.M.,  and  on  June  1st  at  7.30  P.M. 

At  this  hour,  Ursa  Major  is  near  the  zenith,  and  Cassi- 
opeia near  or  below  the  north  horizon.  The  Milky  Way 
is  too  near  the  horizon  to  be  visible.  Orion  has  set  in 
the  west,  and  there  is  no  very  conspicuous  constellation 
in  the  south.  Castor  and  Pollux  are  high  up  in  the 
north-west,  and  Procyon  is  about  an  hour  and  a  half 
above  the  horizon,  a  little  to  the  south  of  west.  All  the 
constellations  in  the  west  and  north-west  have  been  previ- 
ously described,  Leo  being  a  little  west  of  the  meridian. 
Three  zodiacal  constellations  have,  however,  risen,  which 
we  shall  describe. 

Virgo,  the  Virgin,  has  a  single  bright  star,  Spica, 
about  as  bright  as  Regulus,  now  about  one  hour  east  of 
the  meridian,  and  but  little  more  than  half  way  from  the 
zenith  to  the  horizon. 

Libra,  the  Balance,  is  south-east  from  Virgo,  but  has 
no  conspicuous  stars. 


430  ASTRONOMY. 

Scorpius,  the  Scorpion,  is  just  rising  in  the  south-east, 
but  is  not  yet  high  enough  to*  be  well  seen. 

Hydra  is  a  very  long  constellation  extending  from 
Canis  Minor  in  a  south-east  direction  to  the  south  hori- 
zon. Its  brightest  star  is  a  Hydra,  of  the  second  magni- 
tude, 25°  below  Regulus. 

Corvus,  the  Crow,  is  south  of  Virgo,  and  may  be  recog- 
nized by  four  or  five  stars  of  the  second  or  third  magni- 
tude, 15°  south-west  from  Spica. 

Next,  looking  north  of  the  zodiacal  constellations,  we 
see  : 

Coma  Berenices,  the  Hair  of  Berenice,  now  exactly  on 
the  meridian,  and  about  10°  south  of  the  zenith.  It  is  a 
close  irregular  cluster  of  very  small  stars,  unlike  any  thing 
else  in  the  heavens.  In  ancient  mythology,  Berenice  had 
vowed  her  hair  to  Yenus,  but  Jupiter  carried  it  away  from 
the  temple  in  which  it  was  deposited,  and  made  it  into  a 
constellation. 

Bootes,  the  Bear-Keeper,  is  a  large  constellation  east  of 
Coma  Berenices.  It  is  marked  by  Arcturus,  a  bright  but 
somewhat  red  star  of  the  first  magnitude,  about  20°  east 

of  the  zenith.  Bootes  is  repre- 
sented as  holding  two  dogs  in  a 
leash.  These  dogs  are  called 
Canes  Venatici,  and  are  at  the 
time  supposed  exactly  in  our  ze- 
nith chasing  Ursa  Major  around 
the  pole. 

Corona  Borealis,  the  North- 
ern   Crown,    lies   next   east    of 
PIG.   116,-coRONA  BORE-     Bootes  in  the   north-east      It  is 

a  small  but  extremely  beautiful 

constellation.     Its  principal  stars  are  arranged  in  the  form 
of  a  semicircular  chaplet  or  crown. 

Appearance  of  the  Constellations  at  18  Hours  of  Side- 
real Time. — This  hour  occurs  on  July  1st  at  11.30  P.M., 
on  August  1st  at  9.30  P.M.,  and  on  September  1st  at  7.30 
P.M. 


THE  CONSTELLATIONS.  431 

In  this  position,  the  Milky  Way  seems  once  more  to 
span  the  heavens  like  an  arch,  resting  on  the  horizon  in 
the  north-west  and  south-east.  But  we  do  not  see  the 
same  parts  of  it  which  were  visible  in  the  first  position  at 
six  hours  of  right  ascension.  Cassiopeia  is  now  in  the 
north-east  and  Ursa  Major  has  passed  over  to  the 
west. 

Arcturus  is  two  or  three  hours  above  the  western  hori- 
zon. We  shall  commence,  as  in  the  first  position  of  the 
sphere,  by  describing  the  constellations  which  lie  along  on 
the  Milky  Way,  starting  from  Cassiopeia.  Above  Cassi- 
opeia we  have  Cepheus,  and  then  Lacerta,  neither  of 
which  contains  any  striking  stars. 

Cygnus,  the  Swan,  may  be  recognized  by  four  or  five 
stars  forming  a  cross  directly  in  the  centre  of  the  Milky 
Way,  and  a  short  distance  north-east  from  the  zenith. 
The  brightest  of  these  stars,  OL  Cygni,  forms  the  northern 
end  of  the  cross,  and  is  nearly  of  the  first  magnitude. 

Lyra,  the  Harp,  is  a  beautiful  constellation  south-west 
of  Cygnus,  and  nearly  in  the  zenith.  It  contains  the 
brilliant  star  Vega,  or  a 
Lyrce,  of  the  first  mag- 
nitude, and  of  a  bluish 
white  color.  South  of 
Vega  are  four  stars  of 
the  fourth  magnitude, 
forming  an  oblique  par- 
allelogram, by  which  the 
constellation  can  be  read- 
ily recognized.  East  of 
Vega,  and  about  as  far 
from  it  as  the  nearest  FlG'  117--LYRA'  THE  HARP- 
star  of  the  parallelogram,  is  8  Lyrce,  a  very  interesting 
object,  because  it  is  really  composed  of  two  stars  of  the 
fourth  magnitude,  which  can  be  seen  separately  by  a  very 
keen  eye.  The  power  to  see  this  star  double  is  one  of  the 
best  tests  of  the  acuteness  of  one's  vision  (see  Fig.  122). 


432 


ASTRONOMY. 


FlG.    118. — AQUTLA,  DELPHI - 
NUS,  AND    SAGITTA. 


Aquila,  the  Eagle,  is  the  next  striking  constellation  in 
the  Milky  Way.     It  is  two. hours  east  of  the  meridian, 

and  about  midway  between  the 
zenith  and  horizon.  It  is  readily 
recognized  by  the  bright  star 
Altair  or  a  Aquiloe,  situated  be- 
tween two  smaller  ones,  the  one 
of  the  third  and  the  other  of  the 
fourth  magnitude.  The  row  of 
three  stars  lies  in  the  centre  of 
the  Milky  Way. 

Sayitta,  the  Arrow,  is  a  very 
small  constellation,  formed  of 
three  stars  immediately  north  of 
Aquila. 

Delphinus,  the  Dolphin,  is  a 
striking  little  constellation  north-east  of  Aquila,  recog- 
nized by  four  stars  in  the  form  of  a  lozenge.  It  is  famil- 
iarly called  "  Job's  Coffin." 

In  this  position  of  the  celestial  sphere  three  new  zodia- 
cal constellations  have  arisen. 
SCOT  plus,  the  Scorpion, 
already  mentioned,  now  two 
hours  west  of  the  meridian, 
and  about  30°  above  the 
horizon,  is  quite  a  brilliant 
constellation.  It  contains  An- 
tares,  or  a  Scarpii,  a  red- 
dish star  of  nearly  the  first 
magnitude,  and  a  long  row 
of  curved  stars  west  of  it. 

Sagittarius,   the    Archer, 
comprises  a  large  collection 
of  second  magnitude  stars  in  FlG-  H9.— SCORPIUS,  THE   SCOR- 
and   near   the   Milky   Way, 

and  now  very  near  the  meridian.      The  westernmost  stars 
form  the  arrow  of  the  archer. 


THE  CONSTELLATIONS. 


433 


Capricornus,  the  Goat,  is  now  in  the  south-east,  but 
contains  no  bright  stars.  Aquarius,  the  Water-bearer, 
which  has  just  risen,  and  Pisces,  the  Fishes,  which  have 
partly  risen,  contain  no  striking  objects. 

Ophiuchus,  the  Serpent-bearer,  is  a  very  large  constel- 
.ation  north  of  Scorpius  and  west  of  the  Milky  Way. 
Ophiuchus  holds  in  his  hands  an  immense  serpent,  lying 
with  its  tail  in  an  opening  of  the  Milky  Way,  south-west 
of  Aquila,  while  its  head  and  body  are  formed  of  a  col- 
lection of  stars  of  the  third  and  fourth  magnitudes,  ex- 
tending north  of  Scorpius  nearly  to  Bootes. 

Hercules  is  a  very 
large  constellation 
between  Corona 
Borealis  and  Lyra. 
It  is  now  in  the 
zenith,  but  contains 
no  bright  stars.  It 
has,  h  o  w  e  v  e  r  ,  a 
number  of  interest- 
ing telescopic  ob- 
jects, among  them 
the  great  cluster  of 
Hercules,  barely 
visible  to  the  naked 
eye,  but  containing 
an  almost  countless  mass  of  stars.  The  head  of  Draco, 
already  described,  is  just  north  of  Hercules. 

Constellations  Visible  at  0  Hours  of  Sidereal  Time.  — 
This  time  will  occur  on  October  1st  at  11.30  P.M.,  on 
November  1st  at  9.30  P.M.,  on  December  1st  at  7.30  P.M., 
and  on  January  1st  at  5.30  P.M. 

In  this  position,  the  Milky  Way  appears  resting  in  the 
east  and  west  horizons,  but  in  the  zenith  it  is  inclined 
over  toward  the  north.  All  the  constellations,  either  in 
or  north  of  its  course,  are  among  those  already  described. 
We  shall  therefore  consider  only  those  in  the  south. 


FlG.    120.— SAGITTARIUS,    THE   ARCHER. 


434  ASTRONOMY. 

Pegasus,  the  Flying  Horse,  is  distinguished  by  four 
stars  of  the  second  magnitude,  which  form  a  large  square 
about  15°  on  each  side,  called*  the  square  of  Pegasus.  The 
eastern  side  of  this  square  is  almost  exactly  on  the  meri- 
dian. 

Andromeda  is  distinguished  by  a  row  of  three  or  four 
bright  stars,  extending  from  the  north-east  corner  of 
Pegasus,  in  the  direction  of  Perseus. 

Cetus,  the  Whale,  is  a  large  constellation  in  the  south 
and  south-east.  Its  brightest  star  is  /?  Ceti,  standing 
alone,  30°  above  the  horizon,  and  a  little  east  of  the 
meridian. 

Piscis  Australia,  the  Southern  Fish,  lies  further  west 
than  Cetus.  It  has  the  bright  star  Fcrinalhaut,  about 
15C  above  the  horizon,  and  an  hour  west  of  the  meridian. 

§  5.    NUMBERING  AND  CATALOGUING  THE  STARS. 

As  telescopic  power  is  increased,  we  still  find  stars  of 
fainter  and  fainter  light.  But  the  number  cannot  go  on 
increasing  forever  in  the  same  ratio  as  with  the  brighter 
magnitudes,  because,  if  it  did,  the  whole  sky  would  be  a 
blaze  of  starlight. 

If  telescopes  with  powers  far  exceeding  our  present  ones 
were  made,  they  would  no  doubt  show  new  stars  of  the 
20th  and  21st  magnitudes.  But  it  is  highly  probable  that 
the  number  of  such  successive  orders  of  stars  would  not 
increase  in  the  same  ratio  as  is  observed  in  the  8th,  9th, 
and  10th  magnitudes,  for  example.  The  enormous  labor 
of  estimating  the  number  of  stars  of  such  classes  will  long 
prevent  the  accumulation  of  statistics  on  this  question ; 
but  this  much  is  certain,  that  in  special  regions  of  the  sky, 
which  have  been  searchingly  examined  by  various  tele- 
scopes of  successively  increasing  apertures,  the  number  of 
new  stars  found  is  by  no  means  in  proportion  to  the 
increased  instrumental  power.  Thus,  in  the  central  por- 
tions of  the  nebula  of  Orion,  only  some  half  dozen  stars 


CATALOGUING  THE  STARS. 


435 


have  been  found  with  the  Washington  26-inch  refractor 
which  were  not  seen  with  the  Cambridge  15 -inch, 
although  the  visible  magnitude  has  been  extended  from 
15m-l  to  16m>3.  If  this  is  found  to  be  true  elsewhere,  the 
conclusion  may  be  that,  after  all,  the  stellar  system  can  be 
experimentally  shown  to  be  of  finite  extent,  and  to  contain 
only  a  finite  number  of  stars. 

We  have  already  stated  that  in  the  whole  sky  an  eye  of  average 
power  will  see  about  6000  stars.  With  a  telescope  this  number  is 
greatly  increased,  and  the  most  powerful  telescopes  of  modern  times 
will  probably  show  more  than  20,000,000  stars.  As  no  trustworthy 
estimate  has  ever  been  made,  there  is  great  uncertainty  upon  this 
point,  and  the  actual  number  may  range  anywhere  between 
15,000,000  and  40,000,000.  Of  this  number,  not  one  out  of  twenty 
has  ever  been  catalogued  at  all. 

The  gradual  increase  in  the  number  of  stars  laid  down  in  various 
of  the  older  catalogues  is  exhibited  in  the  following  table  from 
CHAMBERS' s  Descriptive  Astronomy  : 


CONSTELLA- 
TION. 

Ptolemy. 
B.C.  130. 

Tycho 
Brahe. 
A.D.  1570. 

Hevelius. 
A.D.  1660. 

Flamsteed. 
A.D.  1690. 

Bode. 

A.D.  1800. 

Aries 

18 

21 

27 

66 

148 

Ursa  Major.. 
Bootes  
Leo 

35 
23 
35 

56 

28 
40 

73 
52 
50 

87 
54 
95 

338 
319 
337 

Virgo  
Taurus  
Orion 

32 
44 
38 

39 
43 
62 

50 
51 
62 

110 
141 

78 

411 
394 
304 

The  most  famous  and  extensive  series  of  star  observations  are 
noticed  below. 

The  uranometries  of  BAYER,  FLAMSTEED,  ARGELANDER,  HEIS,  and 
GOULD  give  the  lucid  stars  of  one  or  both  hemispheres  laid  down 
on  maps.  They  are  supplemented  by  the  star  catalogues  of  other 
observers,  of  which  a  great  number  has  been  published.  These  last 
were  undertaken  mainly  for  the  determination  of  star  positions,  but 
they  usually  give  as  an  auxiliary  datum  the  magnitude  of  the  star 
observed.  When  they  are  carried  so  far  as  to  cover  the  heavens, 
they  will  afford  valuable  data  as  to  the  distribution  of  stars 
throughout  the  sky. 

The  most  complete  catalogue  of  stars  yet  constructed  is  the 
Durchmusterunff  des  Nordlichen  Gestirnten  Himmels,  the  joint  work 
of  ARGELANDER  and  his  assistants,  KRUGER  and  SCHONFELD.  It 
embraces  all  the  stars  of  the  first  nine  magnitudes  from  the  North 


436  ASTRONOMY. 

Pole  to  2°  of  south  declination.  This  work  was  begun  in  1852,  and 
at  its  completion  a  catalogue  of  the  approximate  places  of  no  less 
than  314,926  stars,  with  a  series  of  star-maps,  giving  the  aspect  of 
the  northern  heavens  for  1855,  was*  published  for  the  use  of  astrono- 
mers. ARGELANDER'S  original  plan  was  to  carry  this  Durchmusterung 
as  far  as  23°  south,  so  that  every  star  visible  in  a  small  comet-seeker 
of  2f  inches  aperture  should  be  registered.  His  original  plan  was 
abandoned,  but  his  former  assistant  and  present  successor  at  the 
observatory  of  Bonn,  Dr.  SCHONFELD,  is  now  engaged  in  executing 
this  important  work. 

The  Catalogue  of  Stars  of  the  British  Association  for  the  Ad- 
vancement of  Science  contains  8377  stars  in  both  hemispheres,  and 
gives  all  the  stars  visible  to  the  eye.  It  is  well  adapted  to 
learn  the  unequal  distribution  of  the  lucid  stars  over  the  celestial 
sphere.  The  table  on  the  opposite  page  is  formed  from  its  data. 

From  this  table  it  follows  that  the  southern  sky  has  many  more 
stars  of  the  first  seven  magnitudes  than  the  nort  hern,  and  that  the 
zones  immediately  north  and  south  of  the  Equator,  although  greater 
in  surface  than  any  others  of  the  same  width  in  declination,  are 
absolutely  poorer  in  such  stars. 

The  meaning  of  the  table  will  be  much  better  understood  by  con- 
sulting the  graphical  representation  of  it  on  page  438,  by  PROCTOR. 
On  this  chart  are  laid  down  all  the  stars  of  the  British  Association 
Catalogue  (a  dot  for  each  star),  and  beside  these  the  Milky  Way  is 
represented.  The  relative  richness  of  the  various  zones  can  be  at 
once  seen,  and  perhaps  the  scale  of  the  map  will  allow  the  student 
to  trace  also  the  zone  of  brighter  stars  (lst-3d  magnitude),  which  is 
inclined  to  that  of  the  Milky  Way  by  a  few  degrees,  and  is  approx- 
imately a  great  circle  of  the  sphere. 

The  distribution  and  number  of  the  brighter  stars  (1st- 7th  mag- 
nitude) can  be  well  understood  from  this  chart. 

In  ARGELANDER'S  Durchmusterung  of  the  stars  of  the  northern 
heavens,  there  are  recorded  as  belonging  to  the  northern  hemi- 
sphere : 

10  stars  between  the  1-0  magnitude  and  the  1-9  magnitude. 

37  "  "  2-0  "  "  2-9 

128  "  "  3-0  "  "  3-9 

310  "  "  4-0  "  "  4-9 

1,016  "  "  5-0  "  "  5-9 

4,328  "  "  6-0  "  "  6-9 

13,593  "  "  7-0  '•  "  7-9 

57,960  "  "  8-0  "  "  8-9 

237,544  "  "  9-0  "  "  9-5 

In  all  314,926  stars  from  the  first  to  the  9-5  magnitudes  are  enu- 
merated in  the  northern  sky,  so  that  there  are  about  600,000  in  the 
whole  heavens. 

We  may  readily  compute  the  amount  of  light  received  by  the 
earth  on  a  clear  but  moonless  night  fiom  these  stars.  Let  us  assume 


DISTRIBUTION  OF  STARS. 


437 


4-   + 


\      \ 


i      i 

o?^§ 

O  O 


1        1 


m 

';  x\j&~ 


v..v. 


BRIGHTNESS  OF  THE  STARS.  439 

that  the  brightness  of  an  average  star  of  the  first  magnitude  is 
about  0-5  of  that  of  a  Lyrce.  A  star  of  the  2d  magnitude  will  shine 
with  a  light  expressed  by  0-5  x  0-4= 0-20,  and  so  on. 

The  total  brightness  of       10  1st  magnitude  stars  is    5-0 

37  2d  "  ••  7-4 

122  3d  "  "  10-1 

310  4th  •'  "  9-9 

1,016  5th  '  «  13-0 

4,322  6th  «•  "  22-1 

13,593  7th  ••  «•  27-8 

57,960  8th  "  "  47-4 

Sum  =  142-7 

It  thus  appears  that  from  the  stars  to  the  8th  magnitude,  inclu- 
sive, we  receive  143  times  as  much  light  as  from  a  Lyrce.  a  Lyrce 
has  been  determined  by  ZOLLNER  to  be  about  44,000,000,000  times 
fainter  than  the  sun,  so  that  the  proportion  of  starlight  to  sunlight 
can  be  computed.  It  also  appears  that  the  stars  of  magnitudes  too 
high  to  allow  them  to  be  individually  visible  to  the  naked  eye  are 
yet  so  numerous  as  to  affect  the  general  brightness  of  the  sky  more 
than  the  so-called  lucid  stars  (lst-6th  magnitude). 


CHAPTER    II. 

VARIABLE  AND  TEMPORARY  STARS. 
§  1.  STABS  REGULARLY  VARIABLE. 

ALL  stars  do  not  shine  with  a  constant  light.  Since 
the  middle  of  the  seventeenth  century,  stars  variable  in 
brilliancy  have  been  known,  and  there  are  also  stars  which 
periodically  change  in  color.  The  period  of  a  variable  star 
means  the  interval  of  time  in  which  it  goes  through  all  its 
changes,  and  returns  to  the  same  brilliancy. 

The  most  noted  variable  stars  are  Mira  Ceti  (o  Ceti) 
and  Algol  (ft  Persei).  Mira  appears  about  twelve  times 
in  eleven  years,  and  remains  at  its  greatest  brightness 
(sometimes  as  high  as  the  2d  magnitude,  sometimes  not 
above  the  4th)  for  some  time,  then  gradually  decreases  for 
about  74  days,  until  it  becomes  invisible  to  the  naked  eye, 
and  so  remains  for  about  five  or  six  months.  From  the 
time  of  its  reappearance  as  a  lucid  star  till  the  time  of  its 
maximum  is  about  43  days  (HEIS).  The  mean  period,  or 
the  interval  from  minimum  to  minimum,  is  about  333 
days  (ARGELANDER),  but  this  period,  as  does  the  maxi- 
mum light,  varies  greatly. 

Algol  has  been  known  as  a  variable  star  since  1667.  Its 
period  is  about  2d  20h  49m,  and  is  supposed  to  be  from 
time  to  time  subject  to  slight  fluctuations.  This  star  is 
commonly  of  the  2d  magnitude ;  after  remaining  so 
about  2|  days,  it  falls  to  4m  in  the  short  time  of  4J-  hours, 
and  remains  of  4m  for  20  minutes.  It  then  commences 
to  increase  in  brilliancy,  and  in  another  3£  hours  it  is 


VARIABLE  STARS. 


441 


again  of  the  2d  magnitude,  at  which  point  it  remains  for 
the  remainder  of  its  period,  about  2d  12h. 

These  two  examples  of  the  class  of  variable  stars  give  a 
rough  idea  of  the  extraordinary  nature  of  the  phenomena 
they  present.  A  closer  examination  of  others  discloses 
minor  variations  of  great  complexity  and  apparently  with- 
out law. 

The  following  are  some  of  the  more  prominent  vari- 
able stars  visible  to  the  naked  eye : 


NAME. 

R.  A. 

1870. 

Declination, 
1870. 

Period. 

Changes  of 
Magnitude. 

ft  Persei.     .  .  . 

h.       m.       s. 
2     59     43 

+  40    27-2 

d. 

2-86 

from         to 
2$             4 

(5  Cepliei  

22    24    21 

+  57    45-0 

5-36 

3-7        4-8 

77  Aquilse  

19    45    51 

+    0    40-4 

7-17 

3-6        4-7 

ft  Lyrae 

18    45    17 

-f  33    12-7 

12-91 

3*          4^ 

a  Herculis.  .  .  . 
o   Ceti  
v  Hydrse  
77  Argus  

17      8    43 
2    12    47 
13    22    37 
10    40      2 

+  14    32-4 
-    3    34-1 
-  22    36-4 
-59      0-1 

88-5 
330-0 
436-0 
70  years. 

3-1        3-9 

2          10 
4          10 
1            6 

About  90  variable  stars  are  well  known,  and  as  many 
more  are  suspected  to  vary.  In  nearly  all  cases  the  mean 
period  can  be  fairly  well  determined,  though  anomalies  of 
various  kinds  frequently  appear.  The  principal  anomalies 
are  : 

First.  The  period  is  seldom  constant.  For  some  stars 
the  changes  of  the  period  seem  to  follow  a  regular  law  ; 
for  others  no  law  can  be  fixed. 

Second.  The  time  from  a  minimum  to  the  next  maxi- 
mum is  usually  shorter  than  from  this  maximum  to  the 
next  minimum. 

Third.  Some  stars  (as  ft  Lyrce)  have  not  only  one  max- 
imum between  two  consecutive  principal  minima,  but 
two  such  maxima.  For  ft  Lyrce,  according  to  AKGELAN- 
DER,  3d  2h  after  the  principal  minimum  comes  the  first 
maximum  ;  then,  3d  Th  after  this,  a  secondary  minimum  in 
which  the  star  is  by  no  means  so  faint  as  in  the  principal 


442 


ASTRONOMY. 


minimum,  and  finally  3d  3h  afterward  comes  the  principal 
maximum,  the  whole  period,  being  12d  21h  47'".  The 
course  of  one  period  is  illustrated  below,  supposing  the 
period  to  begin  at  Od  Oh,  and  opposite  each  phase  is  given 
the  intensity  of  light  in  terms  of  y  Lyrce  =  1,  according 
to  photometric  measures  by  KLEIN. 


Phase. 

Relative 
Intensity. 

Principal  Minimum 

Od      Oh 

0-40 

First  Maximum  

3d      2h 

0-83 

Second  Minimum  

6d      9h 

0-58 

Principal  Maximum 

9d     12h 

0-89 

Principal  Minimum.  . 

...  13d    22m 

0-40 

The  periods  of  94r  well-determined  variable  stars  being 
tabulated,  it  appears  that  they  are  as  follows  : 


Period  between 

No.  of  Stars. 

Period  between 

No.  of  Stars. 

Id. 

and    20  d. 

13 

350  d. 

and  400  d. 

13 

20 

50 

1 

400 

450 

8 

50 

100 

4 

450 

500 

3 

100 

150 

4 

500 

550 

0 

150 

200 

5 

550 

600 

0 

200 

250 

9 

600 

650 

1 

250 

300 

14 

650 

700 

0 

300 

350 

18 

700 

750 

1 

2  =94 

It  is  natural  that  there  should  be  few  known  variables 
of  periods  of  500  days  and  over,  but  it  is  not  a  little  re- 
markable that  the  periods  of  over  half  of  these  variables 
should  fall  between  250  and  450  days. 

The  color  of  over  80  per  cent  of  the  variable  stars  is  red 
or  orange.  Red  stars  (of  which  600  to  TOO  are  known) 
are  now  receiving  close  attention,  as  there  is  a  strong  like- 
lihood of  finding  among  them  many  new  variables. 

The  spectra  of  variable  stars  show  changes  which  ap- 
pear to  be  connected  with  the  variations  in  their  light. 


TEMPORAR  Y  STARS.  443 

Another  class  of  variations  occurs  among  the  fixed  stars— namely, 
variations  in  color,  either  with  or  without  corresponding  changes 
of  magnitude. 

In  the  Uranometry,  composed  in  the  middle  of  the  tenth  century 
by  the  Persian  astronomer  AL  SUFI,  it  is  stated  that  at  the  time  of 
his  observations  the  star  Algol  was  reddish — a  term  which  he  ap- 
plies also  to  the  stars  Antares,  Aldebaran,  and  some  others.  Most 
of  these  still  exhibit  a  reddish  aspect.  But  Algol  now  appears  as  a 
white  star,  without  any  sign  of  color.  Dr.  KLEIN,  of  Cologne, 
discovered  that  a  Ursce  Major  is  periodically  changes  color  from  an 
intense  fiery  red  to  a  yellow  or  yellowish-red  every  five  weeks. 
WEBER,  of  Peckeloh,  has  observed  this  star  lately,  and  finds  this 
period  to  be  well  established. 

§  2.    TEMPORARY   OR   NEW  STARS. 

There  are  a  few  cases  known  of  apparently  new  stars 
which  have  suddenly  appeared,  attained  more  or  less 
brightness,  and  slowly  decreased  in  magnitude,  either  dis- 
appearing totally,  or  finally  remaining  as  comparatively 
faint  objects. 

The  most  famous  one  was  that  of  1572,  which  attained 
a  brightness  greater  than  that  of  Sirius  or  Jupiter  and 
approached  to  Venus,  being  even  visible  to  the  eye  in 
daylight.  TYCHO  BRAKE  first  observed  this  star  in  No- 
vember, 1572,  and  watched  its  gradual  increase  in  light 
until  its  maximum  in  December.  It  then  began  to  diminish 
in  brightness,  and  in  January,  1573,  it  was  fainter  than 
Jupiter.  In  February  and  March  it  was  of  the  1st  mag- 
nitude, in  April  and  May  of  the  2d,  in  July  and  August  of 
the  3d,  and  in  October  and  November  of  the  4th.  It  con- 
tinued to  diminish  until  March,  1574,  when  it  became  in- 
visible, as  the  telescope  was  not  then  in  use.  Its  color, 
at  first  intense  white,  decreased  through  yellow  and  red. 
When  it  arrived  at  the  5th  magnitude  its  color  again 
became  white,  and  so  remained  till  its  disappearance. 
TYCHO  measured  its  distance  carefully  from  nine  stars  near 
it,  and  near  its  place  there  is  now  a  star  of  the  10th 
or  llth  magnitude,  which  is  possibly  the  same  star. 

The  history  of  temporary  stars  is  in   general  similar  to 
that  of  the  star  of  1572,  except  that  none  have  attained  so 


444 


ASTRONOMY. 


great  a  degree  of  brilliancy.  More  than  a  score  of  such 
objects  are  known  to  have  appeared,  many  of  them  before 
the  making  of  accurate  observa'tions,  and  the  conclusion  is 
probable  that  many  have  appeared  without  recognition. 
Among  telescopic  stars,  there  is  but  a  small  chance  of  de- 
tecting a  new  or  temporary  star. 

Several  supposed  cases  of  the  disappearance  of  stars  ex- 
ist, but  here  there  are  so  many  possible  sources  of  error 
that  great  caution  is  necessary  in  admitting  them. 

Two  temporary  stars  have  appeared  since  the  invention 
of  the  spectroscope  (1859),  and  the  conclusions  drawn 
from  a  study  of  their  spectra  are  most  important  as  throw- 
ing light  upon  the  phenomena  of  variable  stars  in  general. 

The  first  of  these  stars  is  that  of  1866,  called  T  Coronce. 
It  was  first  seen  on  the  12th  of  May,  1866,  and  was  then 
of  the  2d  magnitude.  Its  changes  were  followed  by  vari- 
ous observers,  and  its  magnitude  found  to  diminish  as 
follows  : 


1866. 

May  12. 
13. 
14, 
15. 
16. 
17. 


m. 
2-0 
2-2 
3-0 
3-5 
4-0 
4-5 


1866.  m. 

May  18 5-5 


19. 
20. 
21. 
22. 
23. 


6-0 
6-5 
7-0 

7-5 
8-0 


By  June  7th  it  had  fallen  to  9ra'0,  and  July  7th  it  was 
9m'5.  SCHMIDT'S  observations  of  this  star  (T  Coronce\ 
continued  up  to  1877,  show  that,  after  falling  from  the 
second  to  the  seventh  magnitude  in  nine  days,  its  light 
diminished  very  gradually  year  after  year  down  to  nearly 
the  tenth  magnitude,  at  which  it  has  remained  pretty  con- 
stant for  some  years.  But  during  the  whole  period  there 
have  been  fluctuations  of  brightness  at  tolerably  regular 
intervals  of  ninety-four  days,  though  of  successively  de- 
creasing extent.  After  the  first  sudden  fall,  there  seems 
to  have  been  an  increase  of  brilliancy,  which  brought  the 
star  above  the  seventh  magnitude  again,  in  October, 
1866,  an  increase  of  a  full  magnitude  ;  but  since  that  time 


VARIABLE  STARS.  445 

the  changes  have  been  much  smaller,  and  are  now  but 
little  more  than  a  tenth  of  a  magnitude.  The  color  of  the 
star  has  been  pale  yellow  throughout  the  whole  course 
of  observations. 

The  spectroscopic  observations  of  this  star  by  HOGGINS  and 
MILLER  showed  it  to  have  a  spectrum  then  absolutely  unique.  The 
report  of  their  observations  says,  "  the  spectrum  of  this  object  is 
twofold,  showing  that  the  light  by  which  it  shines  has  emanated 
from  two  distinct  sources.  The  principal  spectrum  is  analogous 
to  that  of  the  sun,  and  is  formed  of  light  which  was  emitted  by 
an  incandescent  solid  or  liquid  photosphere,  and  which  has  suffered 
a  partial  absorption  by  passing  through  an  atmosphere  of  vapors  at 
a  lower  temperature  than  the  photosphere.  Superposed  over  this 
spectrum  is  a  second  spectrum  consisting  of  a  few  bright  lines 
which  is  due  to  light  which  has  emanated  from  intensely  heated 
matter  in  the  state  of  gas." 

In  November,  1876,  Dr.  SCHMIDT  discovered  a  new  star  in  Cyg- 
nus,  whose  telescopic  history  is  similar  to  that  given  for  T  Cor&nce. 
When  discovered  it  was  of  the  3d  magnitude,  and  it  fell  rapidly 
below  visibility  to  the  naked  eye. 

This  new  star  in  Cygnus  was  observed  by  CORNU,  COPELAND,  and 
VOGEL,  by  means  of  the  spectroscope  ;  and  from  all  the  observa- 
tions it  is  plain  that  the  hydrogen  lines,  at  first  prominent,  have 
gradually  faded.  With  the  decrease  in  their  brilliancy,  a  line 
corresponding  in  position  with  the  brightest  of  the  lines  of  a  nebu- 
la has  strengthened.  On  December  8th,  1876,  this  last  line  was  much 
fainter  than  F  (hydrogen  line  in  the  solar  spectrum),  while  on 
March  3d,  1877,  F  was  very  much  the  fainter  of  the  two. 

At  first  it  exhibited  a  continuous  spectrum  with  numerous  bright 
lines,  but  in  the  latter  part  of  1877  it  emitted  only  monochromatic 
light,  the  spectrum  consisting  of  a  single  bright  line,  correspond- 
ing in  position  to  the  characteristic  line  of  gaseous  nebulaB.  The 
intermediate  stages  were  characterized  by  a  gradual  fading  out, 
not  only  of  the  continuous  spectrum,  but  also  of  the  bright  lines 
which  crossed  it.  From  this  fact,  it  is  inferred  that  this  star,  which 
has  now  fallen  to  10'5  magnitude,  has  actually  become  a  planetary 
nebula,  affording  an  instance  of  a  remarkable  reversal  of  the  pro- 
cess imagined  by  LA  PLACE  in  his  nebular  theory. 


§  3.    THEORIES  OP  VARIABLE  STARS. 

The  theory  of  variable  stars  now  generally  accepted  by  investi- 
gators is  founded  on  the  following  general  conclusions  : 

(1)  That  the  only  distinction  which  can  be  made  between  the 
various  classes  of  stars  we  have  just  described  is  one  of  degree. 
Between  stars  as  regular  as  Algol,  which  goes  through  its  period  in 
less  than  three  days,  and  the  sudden  blazing  out  of  the  star  de- 


446  ASTRONOMY. 

scribed  by  TYCHO  BRAHE,  there  is  every  gradation  of  irregularity. 
The  only  distinction  that  can  be  drawn  between  them  is  in  the 
length  of  the  period  and  the  extent  and  regularity  of  the  changes. 
All  such  stars  must,  therefore,  for  the  present,  be  included  in  the 
single  class  of  variables. 

It  was  at  one  time  supposed  that  newly  created  stars  appeared 
from  time  to  time,  and  that  old  ones  sometimes  disappeared  from 
view.  But  it  is  now  considered  that  there  is  no  well-established 
case  either  of  the  disappearance  of  an  old  star  or  the  creation  of  a 
new  one.  The  supposed  cases  of  disappearance  arose  from  cata- 
loguers accidentally  recording  stars  in  positions  where  none  existed. 
Subsequent  astronomers  finding  no  stars  in  the  place  concluded 
that  the  star  had  vanished  when  in  reality  it  had  never  existed. 
The  view  that  temporary  stars  are  new  creations  is  disproved  by 
the  rapidity  with  which  they  always  fade  away  again. 

(2)  That  all  stars  may  be  to  a  greater  or  less  extent  variable  ; 
only  in  a  vast  majority  of  cases  the  variations  are  so  slight  as  to  be 
imperceptible  to  the  eye.  If  our  sun  could  be  viewed  from  the  dis- 
tance of  a  star,  or  if  we  could  actually  measure  the  amount  of  light 
which  it  transmits  to  our  eyes,  there  is  little  doubt  that  we  should 
tind  it  to  vary  with  the  presence  or  absence  of  spots  on  its  surface. 
We  are  therefore  led  to  the  result  that  variability  of  light  may  be  a 
common  characteristic  of  stars,  and  if  so  we  are  to  look  for  its 
cause  in  something  common  to  all  such  objects. 

The  spots  on  the  sun  may  give  us  a  hint  of  the  probable  cause  of 
the  variations  in  the  light  of  the  stars.  The  general  analogies  of  the 
universe,  and  the  observations  with  the  spectroscope,  all  lead  us  to 
the  conclusion  that  the  physical  constitution  of  the  sun  and  stars  is 
of  the  same  general  nature.  As  we  see  spots  on  the  sun  which  vary 
in  form,  size  and  number  from  day  to  day,  so  if  we  could  take  a  suf- 
ficiently close  view  of  the  faces  of  the  stars  we  should  probably  see 
spots  on  a  great  number  of  them.  In  our  sun  the  spots  never  cover 
more  than  a  very  small  fraction  of  the  surface  ;  but  we  have  no 
reason  to  suppose  that  this  would  be  the  case  with  the  stars.  If 
the  spots  covered  a  large  portion  of  the  surface  of  the  star,  then 
their  variations  in  number  and  extent  would  cause  the  star  to  vary 
in  light. 

This  view  does  not,  however,  account  for  those  cases  in  which  the 
light  of  a  star  is  suddenly  increased  in  amount  hundreds  of  times. 
But  the  spectroscopic  observations  of  T  Corona  show  another 
analogy  with  operations  going  on  in  our  sun.  Mr.  HUGGINS'S  ob- 
servations, which  we  have  already  cited,  seem  to  show  that  there 
was  a  sudden  and  extraordinary  outburst  of  glowing  hydrogen 
from  the  star,  which  by  its  own  light,  as  well  as  by  heating  up  the 
whole  surface  of  the  star,  caused  an  increase  in  its  brilliancy. 

Now,  we  have  on  a  very  small  scale  something  of  this  same  kind 
going  on  in  our  sun.  The  red  flames  which  are  seen  during  a 
total  eclipse  are  caused  by  eruptions  of  hydrogen  from  the  interior 
of  the  sun,  and  these  eruptions  are  generally  connected  with  the 
faculae  or  portions  of  the  sun's  disk  more  brilliant  than  the  rest  of 
the  photosphere. 


VARIABLE  STARS.  447 

The  general  theory  of  variable  stars  which  has  now  the  most 
evidence  in  its  favor  is  this  :  These  bodies  are,  from  some  general 
cause  not  fully  understood,  subject  to  eruptions  of  glowing  hydro- 
gen gas  from  their  interior,  and  to  the  formation  of  dark  spots  on 
their  surfaces.  These  eruptions  and  formations  have  in  most  cases 
a  greater  or  less  tendency  to  a  regular  period. 

In  the  case  of  our  sun,  the  period  is  11  years,  but  in  the  case  of 
many  of  the  stars  it  is  much  shorter.  Ordinarily,  as  in  the  case  of 
the  sun  and  of  a  large  majority  of  the  stars,  the  variations  are  too 
slight  to  affect  the  total  quantity  of  light  to  any  visible  extent. 
But  in  the  case  of  the  variable  stars  this  spot-producing  power  and 
the  liability  to  eruptions  are  very  much  greater  than  in  the  case  of 
our  sun,  and  thus  we  have  changes  of  light  which  can  be  readily 
perceived  by  the  eye.  Some  additional  strength  is  given  to  this 
theory  by  the  fact  just  mentioned,  that  so  large  a  proportion  of 
the  variable  stars  are  red.  It  is  well  known  that  glowing  bodies 
emit  a  larger  proportion  of  red  rays  and  a  smaller  proportion  of 
blue  ones  the  cooler  they  become.  It  is  therefore  probable  that 
the  red  stars  have  the  least  heat.  This  being  the  case,  it  is  more 
easy  to  produce  spots  on  their  surface  ;  and  if  their  outside  surface 
is  so  cool  as  to  become  solid,  the  glowing  hydrogen  from  the  in- 
terior when  it  did  burst  through  would  do  so  with  more  power 
than  if  the  surrounding  shell  were  liquid  or  gaseous. 

There  is,  however,  one  star  of  which  the  variations  may  be  due  to 
an  entirely  different  cause — namely,  Algol.  The  extreme  regularity 
with  which  the  light  of  this  object  fades  away  and  disappears  sug- 
gests the  possibility  that  a  dark  body  may  be  revolving  around  it, 
and  partially  eclipsing  it  at  every  revolution.  The  law  of  variation 
of  its  light  is  so  different  from  that  of  the  light  of  other  variable 
stars  as  to  suggest  a  different  cause.  Most  others  are  near  their 
maximum  for  only  a  small  part  of  their  period,  while  Algol  is  at  its 
maximum  for  nine  tenths  of  it.  Others  are  subject  to  nearly  con- 
tinuous changes,  while  the  light  of  Algol  remains  constant  during 
nine  tenths  of  its  period. 


CHAPTER  III. 

MULTIPLE     STARS. 

§    1.     CHARACTER     OP     DOUBLE    AND    MULTIPLE 

STARS. 

WHEN  we  examine  the  heavens  with  telescopes,  we  find 
many  cases  in  which  two  or  more  stars  are  extremely  close 
together,  so  as  to  form  a  pair,  a  triplet,  or  a  group.  It  is 
evident  that  there  are  two  ways  to  account  for  this  ap- 
pearance. 

1.  We  may  suppose  that  the  stars  happen  to  lie  nearly 
in  the  same  straight  line  from  us,  but  have  no  connection 
with  each  other.     It  is  evident  that  in  this  case  a  pair  of 
stars  might  appear  double,  although  the  one  was  hundreds 
or  thousands  of  times  farther  off  than  the  other.     It  is, 
moreover,  impossible,  from  mere  inspection,  to  determine 
which  is  the  farther. 

2.  We  may  suppose  that  the  stars  are  really  as  near 
together  as  they  appear,  and  are  to  be  considered  as  form- 
ing a  connected  pair  or  group. 

A  couple  of  stars  in  the  first  case  are  said  to  be  optically 
double,  and  are  not  generally  classed  by  astronomers  as 
double  stars. 

Stars  which  are  considered  as  really  double  are  those 
which  are  so  near  together  that  we  are  justified  in  consider- 
ing them  as  physically  connected.  Such  stars  are  said  to 
be  physically  double,  and  are  generally  designated  as 
doitble  stars  simply. 

Though  it  is  impossible  by  mere  inspection  to  decide  to 
which  class  a  pair  of  stars  should  be  considered  as  belong- 
ing, yet  the  calculus  of  probabilities  will  enable  us  to  de- 


DO  UBLE  STARS.  449 

cide  in  a  rough  way  whether  it  is  likely  that  two  stars  not 
physically  connected  should  appear  so  very  close  together 
as  most  of  the  double  stars  do.  This  question  was  first 
considered  by  the  Kev.  JOHN  MICHELL,  F.K.S.,  of  Eng- 
land, who  in  1777  published  a  paper  on  the  subject  in  the 
Philosophical  Transactions.  He  showed  that  if  the  lucid 
stars  were  equally  distributed  over  the  celestial  sphere,  the 
chances  were  80  to  1  against  any  two  being  within  three 
minutes  of  each  other,  and  that  the  chances  were  500,000 
to  1  against  the  six  visible  stars  of  the  Pleiades  being 
accidentally  associated  as  we  see  them.  When  the  mill- 
ions of  telescopic  stars  are  considered,  there  is  a  greater 
probability  of  such  accidental  juxtaposition.  But  the 
probability  of  many  such  cases  occurring  is  so  extremely 
small  that  astronomers  regard  all  the  closest  pairs  as  phy- 
sically connected.  It  is  now  known  that  of  the  600,000 
stars  of  the  first  ten  magnitudes,  at  least  10,000,  or  one  out 
of  every  60,  has  a  companion  within  a  distance  of  30"  of 
arc.  This  proportion  is  many  times  greater  than  could 
possibly  be  the  result  of  chance. 

There  are  several  cases  of  stars  which  appear  double  to 
the  naked  eye.  Two  of  these  we  have  already  described 
— namely,  6  Tauri  and  £  Lyrce.  The  latter  is  a  most 
curious  and  interesting  object,  from  the  fact  that  each  of 
the  two  stars  which  compose  it  is 
itself  double.  No  more  striking 
idea  of  the  power  of  the  teles- 
cope can  be  formed  than  by 
pointing  a  powerful  instrument 
upon  this  object.  It  will  then 
be  seen  that  this  minute  pair  of 
points,  capable  of  being  distin- 
guished only  by  the  most  perfect 
eye,  is  really  composed  of  two  FIG.  122.— THE  QUADRUPLE 
pairs  of  stars  wide  apart,  with  a  STAR  e  LYR^- 

group  of  smaller  stars  between  and  around  them.  The 
figure  shows  the  appearance  in  a  telescope  of  considerable 
power. 


450  ASTRONOMY. 

Revolutions   of  Double   Stars— Binary  Systems. —  The 

most  interesting  question  suggested  by  double  stars  is  that 
of  their  relative  motion.  It  is  evident  that  if  these 
bodies  are  endowed  with  the  property  of  mutual  gravita- 
tion, they  must  be  revolving  around  each  other,  as  the 
earth  and  planets  revolve  around  the  sun,  else  they  would 
be  drawn  together  as  a  single  star.  With  a  view  of  detect- 
ing this  revolution,  astronomers  measure  the  position- 
angle,  and  distance  of  these  objects.  The  distance  of  the 


FlG.    123.— MEASUREMENT  OP  POSITION-ANGLE. 

components  of  the  double  star  is  simply  the  apparent 
angle  which  separates  them,  as  seen  by  the  observer.  It  is 
always  expressed  in  seconds  or  fractions  of  a  second  of  arc. 
The  angle  of  position,  or  "  position-angle"  as  it  is  of  ten 
called  for  brevity,  is  the  angle  which  the  line  joining  the 
two  stars  makes  with  the  line  drawn  from  the  brightest  star 
to  the  north  pole.  If  the  fainter  star  is  directly  north  of 
the  brighter  one,  this  angle  is  zero  ;  if  east,  it  is  90°;  if  south. 


DOUBLE  STARS. 


451 


it  is  180°  ;  if  west,  it  is  270°.  This  is  illustrated  by  tlie 
figure,  which  is  supposed  to  represent  the  field  of  view  of 
an  inverting  telescope  pointed  toward  the  south.  The 
arrow  shows  the  direction  of  the  apparent  diurnal  motion. 
The  telescope  is  supposed  to  be  so  pointed  that  the  brighter 
star  may  be  in  the  centre  of  the  field.  The  numbers 
around  the  surrounding  circle  then  show  the  angle  of  po- 
sition, supposing  the  smaller  star  to  be  in  the  direction  of 
the  number. 

The  letters  sn>  sf,  np,  and  nf  show  the  methods  of 
dividing  the  four  quadrants,  s  meaning  south,  n  north, 
/following,  andj9  preceding.     The  two  latter  words  refer 
to  the  direction  of  the  diur- 
nal motion.     Fig.  124  is  an 
example  of  a  pair  of  stars  in 
which  the  position-angle  is 
about  44°. 

If,  by  measures  of  this 
sort  extending  through  a 
series  of  years,  the  distance 
or  position-angle  of  a  pair 
of  stars  is  found  to  change, 
it  shows  that  one  star  is  re- 
volving around  the  other. 
Such  a  pair  is  called  a 
binary  star  or  binary  sys- 
tem. The  only  distinction 
wThich  we  can  make  between 
binary  systems  and  ordinary  double  stars  is  founded  on 
the  presence  or  absence  of  observed  motion.  It  is  prob- 
able that  nearly  all  the  double  stars  are  really  binary  sys- 
tems, but  that  many  thousands  of  years  are  required  to 
perform  a  revolution,  so  that  the  motion  has  not  yet  been 
detected. 

The  discovery  of  binary  systems  is  one  of  great  scien- 
tific interest,  because  from  them  we  learn  that  the  law  oi 
gravitation  includes  the  stars  as  well  as  the  solar  system  in 


FlG.    124. — POSITION- ANGLE   OF  A 
DOUBLE    STAK. 


452  ASTRONOMY. 

its  scope,  and  may  therefore  be  regarded  as  a  universal 
property  of  matter. 

Colors  of  Double  Stars. — There  are  a  few  noteworthy  statistics 
in  regard  to  the  colors  of  the  components  of  double  stars  which 
may  be  given.  Among  596  of  the  brighter  double  stars,  there  are 
375  pairs  where  each  component  has  the  same  color  and  intensity  ; 
101  pairs  where  the  components  have  same  color,  but  different  in- 
tensity ;  120  pairs  of  different  colors.  Among  those  of  the  same 
color,  the  vast  majority  were  both  white.  Of  the  476  stars  of  the 
same  color,  there  were  295  pairs  whose  components  were  both 
white  ;  118  pairs  whose  components  were  both  yellow  or  both  red  ; 
63  pairs  whose  components  were  both  bluish.  When  the  com- 
ponents are  of  different  colors,  the  brighter  generally  appears  to 
have  a  tinge  of  red  or  yellow  ;  the  other  of  blue  or  green. 

These  data  indicate  in  part  real  physical  laws.  They  also  are 
partly  due  to  the  physiological  fact  that  the  fainter  a  star  is, the 
more  blue  it  will  appear  to  the  eye. 

Measures  of  Double  Stars. — The  first  systematic  measures  of 
the  relative  positions  of  the  components  of  double  stars  were  made 
by  CHRISTIAN  MAYER,  Director  of  the  Ducal  Observatory  of  Mann- 
heim, 1778,  but  it  is  to  SIR  WILLIAM  HERSCIIEL  that  we  owe  the  ba- 
sis of  our  knowledge  of  this  branch  of  sidereal  astronomy.  In  1780 
HERSCHEL  measured  the  relative  situation  of  more  than  400  double 
stars,  and  after  repeating  his  measures  some  score  of  years  later, 
he  found  in  about  50  of  the  pairs  evidence  of  relative  motion  of 
the  components.  In  this  first  survey  he  found  97  stars  whose  dis- 
tance was  under  4",  102  between  4"  and  8",  114  between  8"  and 
16",  and  132  between  16"  and  32". 

Since  HERSCHEL'S  observations,  the  discoveries  of  Sir  JOHN  HER- 
SCHEL, Sir  JAMES  SOUTH,  DAWES,  and  many  others  in  England,  of 
W.  STRUVE,  OTTO  STRUVE,  MADLER,  SECCHI,  DEMBOWSKI,  Du- 
NER,  in  Europe,  and  of  G.  P.  BOND,  ALVAN  CLARK,  and  S.  W. 
BURNHAM,  in  the  United  States,  have  increased  the  number  of 
known  double  stars  to  about  10,000. 

Besides  the  double  stars,  there  are  also  triple,  quadruple,  etc., 
stars.  These  are  generically  called  multiple  stars.  The  most  re- 
markable multiple  star  is  the  Trapezium,  in  the  centre  of  the  nebula 
of  Orion,  commonly  called  Q  Orionis,  whose  four  stars  are,  without 
doubt,  physically  connected. 

The  next  combination  beyond  a  multiple  star  is  a  cluster  of  stars  ; 
and  beginning  with  clusters  of  1'  in  diameter,  such  objects  may  be 
found  up  to  30'  or  more  in  diameter,  every  intermediate  size  being 
represented.  These  we  shall  consider  shortly. 

§  2.     ORBITS  OP  BINARY  STARS. 

When  it  was  established  that  many  of  the  double  stars  were  really 
revolving  around  each  other,  it  became  of  great  interest  to 
determine  the  orbit  and  ascertain  whether  it  was  an  ellipse,  with 


BINARY  STARS. 


453 


the  centre  of  gravity  of  the  two  objects  in  one  of  the  foci  ;  if  so,  it 
would  be  shown  that  gravitation  among  the  stars  followed  the  same 
law  as  in  the  solar  system.  As  an  illustration  of  how  this  may  be 
done,  we  present  the  following  measures  of  the  position-angle  and 
distance  of  the  binary  star  £  Ursce  Majoris,  which  was  the  first  one 
of  which  the  orbit  was  investigated.  The  following  notation  is 
used  :  p,  the  angle  of  position  ;  s,  the  distance  ;  A,  the  brighter 
star  ;  J3,  the  fainter  one. 

f  URSM  MAJORIS  =  2  1523.* 


EPOCH. 

P 

s. 

Observer. 

1782-0 

0 

143-8 

" 

W.  Herschel 

1802  •  1     .              

97-5 

« 

1820-1  

276-4 

W.  Struve. 

1821-8  

264-7 

1-92 

1831-3                   .     ..     . 

201-1 

1-90 

J.  Herschel. 

1840-3     

150-9 

2-45 

Dawes. 

1851-6 

122-6 

2-99 

Madler. 

1863-2 

96-7 

2-56 

Dembowski. 

1872-5       

16-5 

0-91 

Duner. 

If  these  measures  be  plotted  on  a  sheet  of  squared  paper,  the 
several  positions  of  B  will  be  found  to  lie  in  an  ellipse.  This  ellipse 
is  the  projection  of  the  real  orbit  on  the  plane  perpendicular  to  the 
line  of  sight,  or  line  joining  the  earth  with  the  star  A.  It  is  a 
question  of  analysis  to  determine  the  true  orbit  from  the  times  and 
from  the  values  of  p  and  s. 

If  the  real  orbit  happened  to  lie  in  a  plane  perpendicular  to  the 
line  of  sight,  the  star  A  would  lie  in  the  focus  of  the  ellipse.  If 
this  coincidence  does  not  take  place,  then  the  plane  of  the  true  or- 
bit is  seen  obliquely. 

The  first  two  of  KEPLER'S  laws  can  be  employed  in  determining 
such  orbits,  but  the  third  law  is  inapplicable. 

Masses  of  Binary  Systems.— When  the  parallax  or  distance, 
the  semi-major  axis  of  the  orbit,  and  the  time  of  revolution  of  a 
binary  system  are  known,  we  can  determine  the  combined  mass  of 
the  pair  of  stars  in  terms  of  the  mass  of  the  sun.  Let  us  put  : 

a",  the  mean  distance  of  the  two  components  as  measured  in 
seconds  ; 

a,  their  mean  distance  from  each  other  in  astronomical  units  ; 

T,  the  time  of  revolution  in  years  ; 

Mj  J/o,  the  masses  of  the  two  component  stars  ; 

Pj  their  annual  parallax  ; 

J),  their  distance  in  astronomical  units. 


*  2  1523  signifies  that  this  star  is  No.  1523  of  W.  Struve's  Dorpat 
Catalogue. 


454  A8THONOMT. 

From  the  generalization  of  KEPLER'S  third  law,  given  by  the 
theory  of  gravitation,  we  have 


•       y?2  ' 

From  the  formulae  explained  in  treating  of  parallax  we  have 
D  -  I  -4-  sin.  P. 

If  a"  is  the  major  axis  in  seconds,  a  being  the  same  quantity  in 
astronomical  units,  then 

a  =  D  -  sin.  a". 

From  these  two  equations, 

sin.   a"          a" 


sin.  P  '       P 

because  a"  and  P  are  so  small  that  the  arcs  may  be  taken  for  their 
sines. 

Putting  this  value  of  a  in  the  equation  for  M  +  M0, 

a"* 
we  have  M  +  M0  =       mn  ...  • 


a  Centauri  and  p  Ophiuchi  are  two  binary  stars  whose  parallaxes 
have  been  determined  (0"-98  and  0"-16)  from  direct  measures.  For 
a  Centauri 

T—  77-0  years;  a"  =  15" -5;  P  =  0"-98; 

for  p  OpJiiuchi, 

T=  94.4years;  a"  =  4". 70;  P  =  OMG. 

If  we  substitute  in  the  last  equation  these  values  for  T,  P,  and  a", 
we  have 

Mo  +  M  =  0-67  for  a  Centauri, 
Jfo  -I-  M  =  2-84  for;?  Ophiuchi. 

The  last  number  is  quite  uncertain,  owing  to  the  difficulty  of  meas- 
uring so  small  a  parallax.  We  can  only  conclude  that  the  mass  of 
these  two  systems  is  not  many  times  greater  or  less  than  the  mass  of 
our  sun.  From  the  agreement  in  these  two  cases,  it  is  probable  that 
in  other  systems,  if  the  mass  could  be  determined,  it  would  not  be 
greatly  different  from  the  mass  of  our  sun  We  may  on  this  supposi- 
tion, which  amounts  to  supposing  M0  +  M  =  1,  apply  the  formula 

P  =  a"  fTf 

to  other  binaries,  and  deduce  a  value  for  P  in  each  case  which  is  called 
the  hypothetical  parallax  (Gylden),  and  which  is  probably  not  far 
from  the  truth. 

There  are,  beside  binary  systems,  multiple  ones  as  C  Cancri,  where 
the  distance  of  A  and  B  is  0"-8  ;  and  from  the  middle  point  between 

A  and  B  to  C  is  5"  -5.     The  period  of  revolution  of  A  +B  about  C  is 

m 

supposed  to  be  about  730  years.  If  in  the  last  formula  we  put 
T  =  730  years  and  a"  =  5" -5,  we  have  the  hypothetical  parallax 

=  0"-062. 


BINARY  STARS. 


455 


Following  are  given  the  elements  of  several  of  the  more  impor- 
tant binary  stars.  Eight  of  these  have  moved  through  an  entire 
revolution — 360° — since  the  first  observation,  and  about  150  are 
known  which  have  certainly  moved  through  an  arc  of  over  10°  since 
they  were  first  observed. 

In  the  tables  the  semi-major  axis,  or  mean  distance,  must  be 
given  in  seconds,  since  we  have  usually  no  data  by  which  its  value 
in  linear  measures  of  any  kind  can  be  fixed. 

Periods  of  revolution  exceeding  120  years  must  be  regarded  as 
quite  uncertain. 

ELEMENTS  OP  BINARY  STARS. 


STAB'S  NAME. 

Period 
(.Years.) 

Time 
of  Peri- 
astron. 

Semi- 
Axis 
Major. 

Eccen- 
tricity. 

Calculator. 

42  Comse  Ber  
£"  Herculis  

25-7 
34-6 

1869-9 
1864-9 

0"-65 
1-36 

0-48 
0-41 

Dubiago. 
Flammarion 

2  3121*  

37-03 

1842  •  8 

ro-711 

0-26 

Doberck. 

7?  Coronse  Bor.  .  . 
£  Librae  .  .  . 

40-2 
95  90 

1849-9 
1859-6 

0-99 
1-26 

0-29 
0-08 

Flammarion. 
Doberck 

7  Coronse  Aus.  .  . 
£  Ursse  Maj.  .  .  -j 

C  Cancri  ...  .  - 

55-5 
60-6 
60-6 
62-4 

1882-7 
1875.6 
1875-5 
1869-3 

2-40 
2-58 
2-54 
0-90 

0-69 
0-38 
0-37 
0-00 

Schiaparelli. 
Hind. 
Flammarion. 
O.  Struve. 

a.  Centauri  

60-5 
85-0 

1869.9 
1874-9 

0-91 
21-80 

0-37 
0-67 

Flammarion. 
Hind. 

70  Ophiuchi  
y  Coronse  Bor.  .  .  . 
3062  2  

92-8 
95-5 
104-4 

1807-9 
1843-7 
1834-9 

4-88 
0-70 
1-27 

0-39 
0-35 
0-46 

Flammarion. 
Doberck. 
Doberck. 

w  Leonis 

114-6 

1841  •  6 

0-85 

0-55 

Doberck 

Ji  Ophiuchi 

233-9 

1803-9 

1-19 

0-49 

Doberck 

p  Eridani  
1768  2  
£  Bootis  
y  Virginis  
r  Ophiuchi 

117-5 
124-5 
127-4 
175-0 
217-9 

1817-5 
1863-0 
1770-7 
1836-5 
1821-9 

3-82 

4*86 
3-39 
1-40 

0-38 
0-66 
0-71 

0-87 
0-61 

Doberck. 
Doberck. 
Doberck. 
Flammarion. 
Doberck. 

rj  Cassiopeae  
44  Bootis  

222-4 
261-1 

1909-2 
1783-0 

9-83 
3-09 

0-57 
0-71 

Doberck. 
Doberck. 

1938  2  ) 

(j?  Bootis  j 

280-3 

1863-5 

1-47 

0-60 

Doberck. 

36  Andromeda.  .  . 
y  Leonis  
cJ  Cygni  
61  Cygni 

349-1 
402-6 
415-1 
452-0 

1798-8 
1741-1 
1904-1 

1-54 
2-00 
2-31 
15-4 

0-65 
0-74 
0-28 

Doberck. 
Doberck. 
Behrmann. 

a  Coronae  Bor.  .  .  . 
a  Qeminorum.  .  . 
f  Aquarii  

845-9 
1001-2 
1578-3 

1826-9 
1749-8 
1924-2 

5-89 
7-43 
7-64 

0-75 
0-33 
0-65 

Doberck. 
Doberck. 
Doberck. 

*  3121  2  signifies  No.  3121  of  W.  STRUVE'S  Dorpat  Catalogue. 


456  ASTRONOMY. 

The  first  computation  of  the  orbit  of  a  binary  star  was  made  by 
SAVARY  (Astronomer  at  the  Paris  Observatory)  about  1826,  and  his 
results  were  the  first  which  demonstrated  that  the  laws  of  gravita- 
tion, which  we  knew  to  be  operative  over  the  extent  of  the  solar 
system,  and  even  over  the  vast  space  covered  by  the  orbit  of 
H  ALLEY'S  comet,  extended  even  further,  to  the  fixed  stars.  It  might 
have  been  before  1825  a  hazardous  extension  of  our  views  to  sup- 
pose even  the  nearest  fixed  stars  to  be  subject  to  the  laws  of  NEW- 
TON ;  but  as  many  of  the  known  binaries  have  no  measurable  paral- 
lax, it  is  by  no  means  an  unsafe  conclusion  that  every  fixed  star 
which  our  best  telescopes  will  show  is  subjected  to  the  same  laws 
as  those  which  govern  the  fall  of  bodies  upon  the  earth. 


CHAPTER    IV. 

NEBULAE   AND    CLUSTERS. 
§   1.    DISCOVERY  OP  NEBULA. 

IN  the  star-catalogues  of  PTOLEMY,  HEVELIUS  and  the 
earlier  writers,  there  was  included  a  class  of  nebulous  or 
cloudy  stars,  which  were  in  reality  star-clusters.  They 
appeared  to  the  naked  eye  as  masses  of  soft  diffused  light 
of  greater  or  less  extent.  In  this  respect,  they  were  quite 
analogous  to  the  Milky  Way.  When  GALILEO  first  direct- 
ed his  telescope  to  the  sky,  the  nebulous  appearance  of 
these  spots  vanished,  and  they  were  seen  to  consist  of 
clusters  of  stars. 

As  the  telescope  was  improved,  great  numbers  of  such 
patches  of  light  were  found,  some  of  which  could  be  re- 
solved into  stars,  while  others  could  not.  The  latter  were 
called  nebulae  and  the  former  star-clusters. 

About  1650,  HUYGHENS  described  the  great  nebula  of 
Orion,  one  of  the  most  remarkable  and  brilliant  of  these 
objects.  During  the  last  century,  MESSIER,  of  Paris,  made 
a  list  of  103  northern  nebulae,  and  LACAILLE  noted  a  few  of 
those  of  the  southern  sky.  The  careful  sweeps  of  the 
heavens  by  Sir  WILLIAM  HERSCHEL  with  his  great  tele- 
scopes first  gave  proof  of  the  enormous  number  of  these 
masses.  In  1786,  he  published  a  catalogue  of  one  thousand 
new  nebulae  and  clusters.  This  was  followed  in  1789  by 
a  catalogue  of  a  second  thousand,  and  in  1802  by  a  third 
catalogue  of  five  hundred  new  objects  of  this  class.  A 


458  ASTRONOMY. 

similar  series  of  sweeps,  carried  on  by  Sir  JOHN  HEB- 
SCHEL  in  botli  hemispheres,  added  about  two  thousand 
more  nebulae.  The  general  catalogue  of  nebulae  and  clus- 
ters of  stars  of  the  latter  astronomer,  published  in  1864, 
contains  5079  nebulse  :  6251  are  known  in  1879.  Over 
two  thirds  of  these  were  first  discovered  by  the  HEKSCHELS. 
The  mere  enumeration  of  over  4000  nebulae  is,  how- 
ever, but  a  small  part  of  the  labor  done  by  these  two  dis- 
tinguished astronomers.  The  son  has  left  a  great  number 
of  studies,  drawings,  and  measures  of  nebulae,  and  the 
memoirs  of  the  father  on  the  Construction  of  the  'Heavens 
owe  their  suggestiveness  and  much  of  their  value  to  his 
long-continued  observations  on  this  class  of  objects,  which 
gave  him  the  clue  to  his  theories. 

§  2.    CLASSIFICATION  OF  NEBULAS  AND  CLUSTEKS. 

In  studying  these  objects,  the  first  question  we  meet  is 
this  :  Are  all  these  bodies  clusters  of  stars  which  look 
diffused  only  because  they  are  so  distant  that  our  tele- 
scopes cannot  distinguish  them  separately  ?  or  are  some  of 
them  in  reality  what  they  seem  to  be — namely,  diffused 
masses  of  matter  ? 

In  his  early  memoirs  of  1784  and  1785,  Sir  WILLIAM 
HEKSCHEL  took  the  first  view.  He  considered  the  Milky 
Way  as  nothing  but  a  congeries  of  stars,  and  all  nebulae 
naturally  seemed  to  him  to  be  but  stellar  clusters,  so 
distant  as  to  cause  the  individual  stars  to  disappear  in  a 
general  milkiness  or  nebulosity. 

In  1791,  however,  his  views  underwent  a  change.  He 
had  discovered  a  nebulous  star  (properly  so  called),  or  a 
star  which  was  undoubtedly  similar  to  the  surrounding 
stars,  and  which  was  encompassed  by  a  halo  of  nebulous 
light.  * 

*  This  was  the  69th  nebula  of  his  fourth  class  of  planetary  nebulae. 
(H.  iv.  69.) 


NEBULA  AND  CLUSTERS.  459 

He  says  :  "  Nebulae  can  be  selected  so  that  an  insensible  grada- 
tion shall  take  place  from  a  coarse  cluster  like  the  Pleiades  down  to 
a  milky  nebulosity  like  that  in  Orion,  every  intermediate  step  being 
represented.  This  tends  to  confirm  the  hypothesis  that  all  are  com- 
posed of  stars  more  or  less  remote. 

u  A  comparison  of  the  two  extremes  of  the  series,  as  a  coarse 
cluster  and  a  nebulous  star,  indicates,  however,  that  the  nebulosity 
about  the  star  is  not  of  a  starry  nature. 

4 '  Considering  H,  iv.  69,  as  a  typical  nebulous  star,  and  supposing 
the  nucleus  and  chevelure  to  be  connected,  we  may,  first,  suppose 
the  whole  to  be  of  stars,  in  which  case  either  the  nucleus  is  enor- 
mously larger  than  other  stars  of  its  stellar  magnitude,  or  the  envelope 
is  composed  of  stars  indefinitely  small  ;  or,  second,  we  must  admit 
that  the  star  is  involved  in  a  shining  fluid  of  a  nature  totally  unknown 
to  us. 

u  The  shining  fluid  might  exist  independently  of  stars.  The 
light  of  this  fluid  is  no  kind  of  reflection  from  the  star  in  the  cen- 
tre. If  this  matter  is  self-luminous,  it  seems  more  fit  to  produce  a 
star  by  its  condensation  than  to  depend  on  the  star  for  its  existence. 

"  Both  diffused  nebulosities  and  planetary  nebulae  are  better 
accounted  for  by  the  hypothesis  of  a  shining  fluid  than  by  suppos- 
ing them  to  be  distant  stars." 

This  was  the  first  exact  statement  of  the  idea  that,  beside 
stars  and  star-clusters,  we  have  in  the  universe  a  totally 
distinct  series  of  objects,  probably  much  more  simple  in. 
their  constitution.  The  observations  of  HUGGINS  and 
SECOIII  on  the  spectra  of  these  bodies  have,  as  we  shall 
see,  entirely  confirmed  the  conclusions  of  HERSCHEL. 

Nebulae  and  clusters  were  divided  by  HERSCHEL  into 
classes.  Of  his  names,  only  a  few  are  now  in  general  use. 
He  applied  the  name  planetary  nebidcB  to  certain  circular 
or  elliptic  nebulae  which  in  his  telescope  presented  disks 
like  the  planets.  Spiral  nebulae  are  those  whose  convo- 
lutions have  a  spiral  shape.  This  class  is  quite  numer- 
ous. 

The  different  kinds  of  nebula)  and  clusters  will  be  better  under- 
stood from  the  cuts  and  descriptions  which  follow  than  by  formal 
definitions.  It  must  be  remembered  that  there  is  an  almost  infinite 
variety  of  such  shapes. 

The  figure  by  Sir  JOHN  HERSCHEL  on  the  next  page  gives  a  good 
idea  of  a  spiral  or  ring  nebula.  It  has  a  central  nucleus  and  a  small 
and  bright  companion  nebula  near  it.  In  a  larger  telescope  than 
HEUSCIIEL'S  its  aspect  is  even  more  complicated.  See  also  Fig.  128. 


460  ASTRONOMY. 

The  Omega  or  horseshoe  nebula,  so  called  from  the  resemblance 
of  the  brightest  end  of  it  to  a  Greek  Q,  or  to  a  horse's  iron  shoe,  is 
one  of  the  most  complex  and  remarkable  of  the  nebulae  visible  in 
the  northern  hemisphere.  It  is  particularly  worthy  of  note,  as 
there  is  some  reason  to  believe  that  it  has  a  proper  motion.  Cer- 
tain it  is  that  the  bright  star  which  in  the  figure  is  at  the  left-hand 
upper  corner  of  one  of  the  squares,  and  on  the  left-hand  (west) 
edge  of  the  streak  of  nebulosity,  was  in  the  older  drawings  placed 
on  the  other  side  of  this  streak,  or  within  the  dark  bay,  thus  mak- 
ing it  at  least  probable  that  either  the  star  or  the  nebula  has  moved. 


FlG.    125. — SPIRAL  NEBULA. 

The  trifid  nebula,  so  called  on  account  of  its  three  branches 
which  meet  near  a  central  dark  space,  is  a  striking  object,  and 
was  suspected  by  Sir  JOHN  HEHSCHEL  to  have  a  proper  motion. 
Later  observations  seem  to  confirm  this,  and  in  particular  the  three 
bright  stars  on  the  left-hand  edge  of  the  right-hand  (east)  mass  are 
now  more  deeply  immersed  in  the  nebula  than  they  were  observed 
to  be  by  HERSCHEL  (1833)  and  MASON,  of  Yale  College  (1837).  In 
1784,  Sir  WILLIAM  HERSCHEL  described  them  as  "  in  the  middle  of 
the  [dark]  triangle."  This  description  does  not  apply  to  their 
present  situation.  (Fig.  127). 


461 


FlG.  126. — THE  OMEGA  OR  HORSESHOE  NEBULA. 


462 


ASTRONOMY. 


§  3.  STAB  CLUSTERS. 

The  most  noted  of  all  the  clusters  is  the  Pleiades,  which  have 
already  been  briefly  described  in  connection  with  the  constellation 
Taurus.  The  average  naked  eye  can  easily  distinguish  six  stars 
within  it,  but  under  favorable  conditions  ten,  eleven,  twelve,  or 


PlG.    127. — THE  TRIFID  NEBULA. 

more  stars  can  be  counted.  With  the  telescope,  over  a  hundred 
stars  are  seen.  A  view  of  these  is  given  in  the  map  accompanying 
the  description  of  the  Pleiades,  Fig.  113,  p.  425.  This  group  con- 
tains TEMPEL'S  variable  nebula,  so  called  because  it  has  been  sup- 
posed to  be  subject  to  variations  of  light.  This  is  probably  not  a 
variable  nebula. 


NEBULAE  AND  CLUSTERS. 


463 


The  clusters  represented  in  Figs.  129  and  130  are  good  examples 
of  their  classes.  The  first  is  globular  and  contains  several  thousand 
small  stars.  The  central  regions  are  densely  packed  with  stars, 
and  from  these  radiate  curved  hairy-looking  branches  of  a  spiral 
form.  The  second  is  a  cluster  of  about  200  stars,  of  magnitudes 
varying  from  the  ninth  to  the  thirteenth  and  fourteenth,  in  which 
the  brighter  stars  are  scattered  in  a  somewhat  unusual  manner 


FlG.  128. — THE  RING  NEBULA  IN  LYRA. 

over  the  telescopic  field.  This  cluster  is  an  excellent  example  of 
the  "  compressed  "  form  so  frequently  exhibited.  In  clusters  of 
this  class  the  spectroscope  shows  that  each  of  the  individual  stars 
is  a  true  sun,  shining  by  its  native  brightness.  If  we  admit  that  a 
cluster  is  real — that  is,  that  we  have  to  do  with  a  collection  of  stars 
physically  connected — the  globular  clusters  become  important.  It 
is  a  fact  of  observation  that  in  general  the  stars  composing  such 


464 


ASTRONOMY. 


clusters  are  about  of  equal  magnitude,  and  are  more  condensed  at 
the  centre  than  at  the  edges.  They  are  probably  subject  to  central 
powers  or  forces.  This  was  seen  by  Sir  WILLIAM  HERSCHEL  in  1789. 
He  says  : 

"  Not  only  were  round  nebulae  and  clusters  formed  by  central 
powers,  but  likewise  every  cluster  of  stars  or  nebula  that  shows  a 
gradual  condensation  or  increasing  brightness  toward  a  centre. 
This  theory  of  central  power  is  fully  established  on  grounds  of  ob- 
servation which  cannot  be  overturned. 

' '  Clusters  can  be  found  of  1 0  diameter  with  a  certain  degree  of 
compression  and  stars  of  a  certain  magnitude,  and  smaller  clusters 
of  4',  3'  or  2'  in  diameter,  with  smaller  stars  and  greater  compression, 
and  so  on  through  resolvable  nebulae  by  imperceptible  steps,  to  the 
smallest  and  faintest  [and  most  distant]  nebulae.  Other  clusters 


FlO.    120. — GLOBULAR  CLUSTER.          FlG.    130. — COMPRESSED   CLUSTER. 


there  are,  which  lead  to  the  belief  that  either  they  are  more  com- 
pressed or  are  composed  of  larger  stars.  Spherical  clusters  are 
probably  not  more  different  in  size  among  themselves  than  different 
individuals  of  plants  of  the  same  species.  As  it  has  been  shown 
that  the  spherical  figure  of  a  cluster  of  stars  is  owing  to  central 
powers,  it  follows  that  those  clusters  which,  cmteris  parilus,  are  the 
most  complete  in  this  figure  must  have  been  the  longest  exposed 
to  the  action  of  these  causes. 

"  The  maturity  of  a  sidereal  system  may  thus  be  judged  from 
the  disposition  of  the  component  parts. 

"  Though  we  cannot  see  any  individual  nebula  pass  through  all 
its  stages  of  life,  we  can  select  particular  ones  in  each  peculiar 
stage,"  and  thus  obtain  a  single  view  of  their  entire  course  of  de- 
velopment. 


NEBULAS.  405 


§  4.    SPECTRA  OP  NEBULAE  AND  CLUSTERS. 

In  18G4,  five  years  after  the  invention  of  the  spectroscope,  Dr. 
HUGGINS,  of  London,  commenced  the  examination  of  the  spectra 
of  the  nebulae,  and  was  led  to  the  discovery  that  while  the  spectra 
of  stars  were  invariably  continuous  and  crossed  with  dark  lines 
similar  to  those  of  the  solar  spectrum,  those  of  many  nebulae  were 
discontinuous,  showing  these  bodies  to  be  composed  of  glowing  gas. 
The  figure  shows  the  spectrum  of  one  of  the  most  famous  planetary 
nebulae.  (H.  iv.  37.)  The  gaseous  nebulae  include  nearly  all  tho 
planetary  nebulae,  and  very  frequently  have  stellar-like  condensa- 
tions in  the  centre. 

Singular  enough,  the  most  milky  looking  of  any  of  the  nebula? 
(that  in  Andromeda)  gives  a  continuous  spectrum,  while  the  nebula 
of  Orion,  which  fairly  glistens  with  small  stars,  has  a  discontinuous 


-Ba 


FlG.    131. — SPECTRUM   OP  A  PLANETARY  NEBULA. 

spectrum,  showing  it  to  be  a  true  gas.  Most  of  these  stars  are  too 
faint  to  be  separately  examined  with  the  spectroscope,  so  that  we 
cannot  say  whether  they  have  the  same  spectrum  as  the  nebulae. 

The  spectrum  of  most  clusters  is  continuous,  indicating  that  the 
individual  stars  are  truly  stellar  in  their  nature.  In  a  few  cases, 
however,  clusters  are  composed  of  a  mixture  of  nebulosity  (usually 
near  their  centre)  and  of  stars,  and  the  spectrum  in  such  cases  is 
compound  in  its  nature,  so  as  to  indicate  radiation  both  by  gaseous 
and  solid  matter. 


§   5.    DISTRIBUTION  OP  NEBULA  AND   CLUSTERS 
ON  THE  SURFACE  OF  THE   CELES- 
TIAL SPHERE. 

The  following  map  (Fig.  132)  by  Mr.  R.  A.  PROCTOR,  gives  at  a 
glance  the  distribution  of  the  nebulae  on  the  celestial  sphere  with 
reference  to  the  Milky  Way,  whose  boundaries  only  are  indicated. 


STAR-CL  USTERS.  467 

The  position  of  each  nebula  is  marked  by  a  dot ;  where  the  dots  are 
thickest  there  is  a  region  rich  in  nebula?.  A  casual  examination 
shows  that  such  rich  regions  are  distant  from  the  Galaxy,  and  it 
would  appear  that  it  is  a  general  law  that  the  nebulae  are  distri- 
buted in  greatest  number  around  the  two  poles  of  the  galactic 
circle,  and  that  in  a  general  way  their  number  at  any  point  of  the 
sphere  increases  with  their  distance  from  this  circle.  This  was 
noticed  by  the  elder  HERSCHEL,  who  constructed  a  map  similar  to 
the  one  given.  It  is  precisely  the  reverse  of  the  law  of  apparent 
distribution  of  the  true  star-clusters,  which  in  general  lie  in  or  near 
the  Milky  Way. 


CHAPTER    V. 

SPECTRA   OF   FIXED    STARS. 

1.    CHARACTERS  OP    STELLAR  SPECTRA. 

SOON  after  the  discovery  of  the  spectroscope,  Dr.  HUGGINS  and 
Professor  W.  A.  MILLER  applied  this  instrument  to  the  examina- 
tion of  stellar  spectra,  which  were  found  to  be,  in  the  main,  similar 
to  the  solar  spectrum — i.e.,  composed  of  a  continuous  band  of  the 
prismatic  colors,  across  which  dark  lines  or  bands  were  laid,  the 
latter  being  fixed  in  position.  These  results  showed  the  fixed  stars 
to  resemble  our  own  sun  in  general  constitution,  and  to  be  com- 
posed of  an  incandescent  nucleus  surrounded  by  a  gaseous  and 
absorptive  atmosphere  of  lower  temperature.  This  atmosphere 
around  many  stars  is  different  in  constitution  from  that  of  the  sun, 
as  is  shown  by  the  different  position  and  intensity  of  the  various 
black  lines  and  bands. 

The  various  stellar  spectra  have  been  classified  by  SECCIII  into 
four  type*,  distinguished  from  one  another  by  marked  differences  in 
the  position,  character,  and  number  of  the  dark  lines. 

Type  I  is  composed,  of  the  white  stars,  of  which  Sirius  and  Vcya 
are  examples  (the  upper  spectrum  in  the  plate  Fig.  133).  The  spec- 
trum of  these  stars  is  continuous,  and  is  crossed  by  four  dark 
lines,  due  to  the  presence  of  large  quantities  of  hydrogen  in 
the  envelope.  Sodium  and  magnesium  lines  are  also  seen,  and 
others  yet  fainter. 

Type  II  is  composed  mainly  of  the  yellow  stars,  like  our  own 
sun,  Arcturus,  Capella,  Aldebaran,  and  Pollux.  The  spectrum  of 
the  sun  is  shown  in  the  second  place  in  the  plate.  The  vast  ma- 
jority of  the  stars  visible  to  the  naked  eye  belong  to  this  class. 

Type  III  (see  the  third  and  fourth  spectra  in  the  plate)  is  com- 
posed of  the  brighter  reddish  stars  like  a  Orionis,  Antares,  a  Ilerculis, 
etc.  These  spectra  are  much  contracted  toward  the  violet  end,  and 
are  crossed  by  eight  or  more  dark  bands,  these  bands  being  them- 
selves resolvable  into  separate  lines. 

These  three  types  comprise  nearly  all  the  lucid  stars,  and  it  is 
not  a  little  remarkable  that  the  essential  differences  between  the 
three  classes  were  recognized  by  Sir  WILLIAM  HERSCHEL  as  early 
as  1798,  and  published  in  1814.  Of  course  his  observations  were 
made  without  a  slit  to  his  spectroscopic  apparatus. 


8TKLLAR  SPECTRA. 


469 


470  ASTRONOMY. 

Type  IV  comprises  the  red  stars,  which  are  mostly  telescopic. 
The  characteristic  spectrum  is  shown  in  the  last  figure  of  the  plate. 
It  is  curiously  banded  with  three  bright  spaces  separated  by 
darker  ones. 

It  is  probable  that  the  hotter  a  star  is  the  more  simple  a  spectrum 
it  has  ;  for  the  brightest,  and  therefore  probably  the  hottest  stars, 
such  as  SiriuSj  give  spectra  showing  only  very  thick  hydrogen  lines 
and  a  few  very  thin  metallic  lines,  while  the  cooler  stars,  such  as 
our  sun,  are  shown  by  their  spectra  to  contain  a  much  larger  num- 
ber of  metallic  elements  than  stars  of  the  type  of  Sirius,  but  no 
non-metallic  elements  (oxygen  possibly  excepted).  The  coolest 
stars  give  band-spectra  characteristic  of  compounds  of  metallic 
with  non-metallic  elements,  and  of  the  non-metallic  elements  un- 
combined. 


§  2.    MOTION  OP  STABS  IN  THE  LINE  OF  SIGHT. 

Spectroscopic  observations  of  stars  not  only  give  information  in 
regard  to  their  chemical  and  physical  constitution,  but  have  been 
applied  so  as  to  determine  approximately  the  velocity  in  kilometres 
per  second  with  which  the  stars  are  approaching  to  or  receding 
from  the  earth  along  the  line  joining  earth  and  star.  The  theory 
of  such  a  determination  is  briefly  as  follows  : 

In  the  solar  spectrum  we  find  a  group  of  dark  lines,  as  «,  &,  c, 
which  always  maintain  their  relative  position.  From  laboratory 
experiments,  we  can  show  that  the  three  bright  lines  of  incandescent 
hydrogen  (for  example)  have  always  the  same  relative  position  as 
the  solar  dark  lines  a,  &,  c.  From  this  it  is  inferred  that  the  solar 
dark  lines  are  due  to  the  presence  of  hydrogen  in  its  absorptive 
atmosphere. 

Now,  suppose  that  in  a  stellar  spectrum  we  find  three  dark 
lines  a't  &',  c',  whose  relative  position  is  exactly  the  same  as  that 
of  the  solar  lines  a,  ft,  c.  Not  only  is  their  relative  position  the 
same,  but  the  characters  of  the  lines  themselves,  so  far  as  the  fainter 
spectrum  of  the  star  will  allow  us  to  determine  them,  are  also  simi- 
lar— that  is,  a'  and  «,  &'  and  ft,  c'  and  c  are  alike  as  to  thickness, 
blackness,  nebulosity  of  edges,  etc. ,  etc.  From  this  it  is  inferred 
that  the  star  really  contains  in  its  atmosphere  the  substance  whose 
existence  has  been  shown  in  the  sun. 

If  we  contrive  an  apparatus  by  which  the  stellar  spectrum  is  seen 
in  the  lower  half  (say)  of  the  eye-piece  of  the  spectroscope,  while 
the  spectrum  of  hydrogen  is  seen  just  above  it,  we  find  in  some 
cases  this  remarkable  phenomenon.  The  three  dark  stellar  lines, 
a',  &',  c',  instead  of  being  exactly  coincident  with  the  three  hydro- 
gen lines  a,  &,  c ,  are  seen  to  be  all  thrown  to  one  side  or  the 
other  by  a  like  amount— that  is,  the  whole  group  a',  &',  c',  while 
preserving  its  relative  distances  the  same  as  those  of  the  compari- 
son group  a,  &,  c,  is  shifted  toward  either  the  violet  or  red  end  of 
the  spectrum  by  a  small  yet  measurable  amount.  Repeated  experi- 


STELLAR  SPECTRA. 


471 


mcnts  by  different  instruments  and  observers  show  always  a  shifting 
in  the  same  direction  and  of  like  amount.  The  figure  shows  the 
shifting  of  the  F  line  in  the  spectrum  of  fiiriw,  compared  with  one 
fixed  line  of  hydrogen. 

This  displacement  of  the 
spectral  lines  is  now  ac- 
counted for  by  a  motion  of 
the  star  toward  or  from  the 
earth.  It  is  shown  in  Phy- 
sics that  if  the  source  of 
the  light  which  gives  the 
spectrum  #',  &',  c'  is  mov- 
ing away  from  the  earth,  this 
group  will  be  shifted  toward 
the  red  end  of  the  spec- 
trum ;  if  toward,  the  earth, 
then  the  whole  group  will 
be  shifted  toward  the  blue 
end.  The  amount  of  this 
shifting  is  a  function  of  the 
velocity  of  recession  or  ap- 
proach, and  this  velocity  in 
miles  per  second  can  be 
calculated  from  the  meas- 
ured displacement.  This  has  been  done  for  many  stars  by  Dr. 
HUGGINS,  Dr.  VOGEL,  and  Mr.  CIIIUSTIE.  Their  results  agree  well, 
when  the  difficult  nature  of  the  research  is  considered.  The  rates 
of  motion  vary  from  insensible  amounts  to  100  kilometres  per  sec- 
ond ;  and  in  some  cases  agree  remarkably  with  the  velocities  com- 
puted from  the  proper  motions  and  probable  parallaxes. 


FlG.    134. — F-LTNK   IN   SPECTRUM   OF 
SIRIUS. 


CHAPTER   VI. 

MOTIONS   AND    DISTANCES   OF   THE   STARS. 
§    1.    PROPER  MOTIONS. 

WE  have  already  stated  that,  to  the  unaided  vision,  the 
fixed  stars  appear  to  preserve  the  same  relative  position  in 
the  heavens  through  many  centuries,  so  that  if  the  an- 
cient astronomers  once  more  saw  them,  they  could  hardly 
detect  the  slightest  change  in  their  arrangement.  But 
the  refined  methods  of  modern  astronomy,  in  which  the 
power  of  the  telescope  is  applied  to  celestial  measurement, 
have  shown  that  there  are  slow  changes  in  the  positions 
of  the  brighter  stars,  consisting  in  a  motion  forward  in  a 
straight  line  and  with  uniform  velocity.  These  motions 
are,  for  the  most  part,  so  slow  that  it  would  require  thou- 
sands of  years  for  the  change  of  position  to  be  percepti- 
ble to  the  unaided  eye.  They  are  called  proper  motions. 

As  a  general  rule,  the  fainter  the  stars  the  smaller  the  proper  mo- 
tions. For  the  most  part,  the  proper  motions  of  the  telescopic  stars 
are  so  minute  that  they  have  not  been  detected  except  in  a  very 
few  cases.  This  arises  partly  from  the  actual  slowness  of  the  mo- 
tion, and  partly  from  the  fact  that  the  positions  of  these  stars  have 
not  generally  been  well  determined.  It  will  be  readily  seen  that,  in 
order  to  detect  the  proper  motion  of  a  star,  its  position  must  be  de- 
termined at  periods  separated  by  considerable  intervals  of  time. 
Since  the  exact  determinations  of  star  positions  have  only  been 
made  since  the  year  1750,  it  follows  that  no  proper  motion  can  be 
detected  unless  it  is  large  enough  to  become  perceptible  at  the  end 
of  a  century  and  a  quarter.  With  very  few  exceptions,  no  accurate 
determination  of  the  positions  of  telescopic  stars  was  made  until 
about  the  beginning  of  the  present  century.  Consequently,  we 
cannot  yet  pronounce  upon  the  proper  motions  of  these  stars,  and 


MOTIONS  OF  THE  STARS.  473 

can  only  say  that,  in  general,  they  are  too  small  to  be  detected  by 
the  observations  hitherto  made. 

To  this  rule,  that  the  smaller  stars  have  no  sensible  proper  mo- 
tions, there  are  a  few  very  notable  exceptions.  The  star  Groom- 
bridge  1830,  is  remarkable  for  having  the  greatest  proper  motion  of 
any  in  the  heavens,  amounting  to  about  7"  in  a  year.  It  is  only  of 
the  seventh  magnitude.  Next  in  the  order  of  proper  motion  comes 
the  double  star  61  Cygni,  which  is  about  of  the  fifth  magnitude. 
There  are  in  all  seven  small  stars,  all  of  which  have  a  larger  proper 
motion  than  any  of  the  first  magnitude.  But  leaving  out  these  ex- 
ceptional cases,  the  remaining  stars  show,  on  an  average,  a  diminu- 
tion of  proper  motion  with  brightness.  In  general,  the  proper 
motions  even  of  the  brightest  stars  are  only  a  fraction  of  a  second 
in  a  year,  so  that  thousands  of  years  would  be  required  for  them 
to  change  their  place  in  any  striking  degree,  and  hundreds  of 
thousands  to  make  a  complete  revolution  around  the  heavens. 


§  2.    PROPER  MOTION  OF  THE  SUN. 

A  very  interesting  result  of  the  proper  motions  of  the 
stars  is  that  our  sun,  considered  as  a  star,  has  a  consider- 
able proper  motion  of  its  own.  By  observations  on  a  star, 
we  really  determine,  not  the  proper  motion  of  the  star  it- 
self, but  the  relative  proper  motion  of  the  observer  and 
the  star — that  is,  the  difference  of  their  motions.  Since 
the  earth  with  the  observer  on  it  is  carried  along  with  the 
sun  in  space,  his  proper  motion  is  the  same  as  that  of  the 
sun,  so  that  what  observation  gives  us  is  the  difference 
between  the  proper  motion  of  the  star  and  that  of  the  sun. 
There  is  no  way  to  determine  absolutely  how  much  of 
the  apparent  proper  motion  is  due  to  the  real  motion  of 
the  star  and  how  much  to  the  real  motion  of  the  sun.  If, 
however,  we  find  that,  on  the  average,  there  is  a  large  pre- 
ponderance of  proper  motions  in  one  direction,  we  may 
conclude  that  there  is  a  real  motion  of  the  sun  in  an  op- 
posite direction.  The  reason  of  this  is  that  it  is  more 
likely  that  the  average  of  a  great  mass  of  stars  is  at  rest 
than  that  the  sun,  which  is  only  a  single  one,  should  be  at 
rest.  Now,  observation  shows  that  this  is  really  the  case, 
and  that  the  great  mass  of  stars  appear  to  be  moving  from 
the  direction  of  the  constellation  Hercules  and  toward 


474 


ASTRONOMY. 


that  of  the  constellation  Argus.*  A  number  of  astrono- 
mers have  investigated  this  motion  with  a  view  of  deter- 
mining the  exact  point  in  the  heavens  toward  which  the 
sun  is  moving.  Their  results  are  shown  in  the  following 
table  : 


Right  Ascension. 

Declination. 

Arfelander  

257° 
261° 
252° 
260° 
261° 
2G2° 

49' 
22' 
24' 

1' 
38' 
29' 

28°    50'    N. 
37°    36'    N. 
14°    26'    N. 
34°    23'    N. 
39°     54'     N. 
28°    58'    N. 

O.  Struve  

Lundabl 

Galloway  ... 

Miidler  

Airy  and  Dunkin  

It  will  be  perceived  that  there  is  some  discordance  aris- 
ing from  the  diverse  characters  of  the  motions  to  be  in- 
vestigated. Yet,  if  we  lay  these  different  points  down  on 
a  map  of  the  stars,  we  shall  find  that  they  all  fall  in  the 
constellation  Hercules.  The  amount  of  the  motion  is  such 
that  if  the  sun  were  viewed  at  right  angles  to  the  direction 
of  motion  from  an  average  star  of  the  first  magnitude,  it 
would  appear  to  move  about  one  third  of  a  second  per 
year. 

§   3.    DISTANCES  OP  THE  FIXED  STARS. 

The  problem  of  the  distance  of  the  stars  has  always 
been  one  of  the  greatest  interest  on  account  of  its  involv- 
ing the  question  of  the  extent  of  the  visible  universe. 
The  ancient  astronomers  supposed  all  the  fixed  stars  to  be 
situated  at  a  short  distance  outside  of  the  orbit  of  the  planet 
Saturn,  then  the  outermost  known  planet.  The  idea  was 
prevalent  that  Nature  would  not  waste  space  by  leaving  a 
great  region  beyond  /Saturn  entirely  empty. 

When  COPERNICUS  announced  the  theory  that  the  sun 
was  at  rest  and  the  earth  in  motion  around  it,  the  prob- 
lem of  the  distance  of  the  stars  acquired  a  new  interest. 

*  This  was  discovered  by  Sir  WILLIAM  HERSCHEL  in  1783. 


DISTANCES  OF  THE  STARS.  475 

It  was  evident  that  if  the  earth  described  an  annual  orbit, 
then  the  stars  would  appear  in  the  course  of  a  year  to  os- 
cillate back  and  forth  in  corresponding  orbits,  unless  they 
were  so  immensely  distant  that  these  oscillations  were  too 
small  to  be  seen.  Now,  the  apparent  oscillation  of  Saturn 
produced  in  this  way  was  described  in  Part  I. ,  and  shown 
to  amount  to  some  6°  on  each  side  of  the  mean  position. 
These  oscillations  were,  in  fact,  those  which  the  ancients 
represented  by  the  motion  of  the  planet  around  a  small 
epicycle.  But  no  such  oscillation  had  ever  been  detected 
in  a  fixed  star.  This  fact  seemed  to  present  an  almost 
insuperable  difficulty  in  the  reception  of  the  Copernican 
system.  This  was  probably  the  reason  why  TYCHO  BBAIIE 
was  led  to  reject  the  system.  Yery  naturally,  therefore, 
as  the  instruments  of  observation  were  from  time  to  time 
improved,  this  apparent  annual  oscillation  of  the  stars  was 
ardently  sought  for.  When,  about  the  year  1704, 
EOEMER  thought  he  had  detected  it,  he  published  his  ob- 
servations in  a  dissertation  entitled  "  Copernicus  Trium- 
phans"  A  similar  attempt,  made  by  HOOKE  of  England, 
was  entitled  "  An  Attempt  to  Prove  the  Motion  of  the 
Earth." 

Tliis  problem  is  identical  with  that  of  the  annual  paral- 
lax of  the  fixed  stars,  which  has  been  already  described  in 
the  concluding  section  of  our  opening  chapter.  This 
parallax  of  a  heavenly  body  is  the  angle  which  the  mean 
distance  of  the  earth  from  the  sun  subtends  when  seen 
from  the  body.  The  distance  of  the  body  from  the  sun  is 
inversely  as  the  parallax  (nearly).  Thus  the  mean  distance 
of  Saturn  being  9  •  5,  its  annual  parallax  exceeds  6°,  while 
that  of  Neptune,  which  is  three  times  as  far,  is  about  2°. 
It  was  very  evident,  without  telescopic  observation,  that 
the  stars  could  not  have  a  parallax  of  one  half  a  degree. 
They  must  therefore  be  at  least  twelve  times  as  far  as 
Saturn  if  the  Copernican  system  were  true. 

When  the  telescope  was  applied  to  measurement,  a  con- 
tinually increasing  accuracy  began  to  be  gained  by  the 


476  ASTRONOMY. 

improvement  of  the  instruments.  Yet  for  several  genera- 
tions the  parallax  of  the  fixed^  stars  eluded  measurement. 
Very  often  indeed  did  observers  think  they  had  detected 
a  parallax  in  some  of  the  brighter  stars,  but  their  succes- 
sors, on  repeating  their  measures  with  better  instruments, 
and  investigating  their  methods  anew,  found  their  con- 
clusions erroneous.  Early  in  the  present  century  it  be- 
came certain  that  even  the  brighter  stars  had  not,  in  gen- 
eral, a  parallax  as  great  as  1",  and  thus  it  became  certain 
that  they  must  lie  at  a  greater  distance  than  200,000  times 
that  which  separates  the  earth  from  the  sun. 

Success  in  actually  measuring  the  parallax  of  the  stars 
was  at  length  obtained  almost  simultaneously  by  two  as- 
tronomers, BEBSEL  of  Konigsberg,  and  STRUVE  of  Dorpat. 
BESSEL  selected  for  his  star  to  be  observed  61  Cyyn-i,  and 
commenced  his  observations  on  it  in  August,  1837.  The 
result  of  two  or  three  years  of  observation  was  that  this 
star  had  a  parallax  of  0"-35,  or  about  one  third  of  a  sec- 
ond. This  would  make  its  distance  from  the  sun  nearly 
600,000  astronomical  units.  The  reality  of  this  paral- 
lax has  been  well  established  by  subsequent  investigators, 
only  it  has  been  shown  to  be  a  little  larger,  and  therefore 
the  star  a  little  nearer  than  BESSEL  supposed.  The  most 
probable  parallax  is  now 'found  to  be  0"-51,  corresponding 
to  a  distance  of  400,000  radii  of  the  earth's  orbit. 

The  star  selected  by  STRUVE  for  the  measure  of  parallax  was  the 
bright  one,  a  Lyrce.  His  observations  were  made  between  Novem- 
ber, 1835,  and  August,  1838.  He  first  deduced  a  parallax  of  0"-25. 
Subsequent  observers  have  reduced  this  parallax  to  0"  •  20,  corre- 
sponding to  a  distance  of  about  1,000,000  astronomical  units. 

Shortly  after  this,  it  was  found  by  HENDERSON,  of  England,  As- 
tronomer Royal  for  the  Cape  of  Good  Hope,  that  the  star  a  Ceutntiri 
had  a  still  larger  parallax  of  about  1".  This  is  the  largest  parallax 
now  known  in  the  case  of  any  fixed  star,  so  that  a  Centauri  is,  be- 
yond all  reasonable  doubt,  the  nearest  fixed  star.  Yet  its  distance 
is  more  than  200,000  astronomical  units,  or  thirty  millions  of  mill- 
ions of  kilometres.  Light,  which  passes  from  the  sun  to  the  earth 
in  8  minutes,  would  require  3£  years  to  reach  us  from  a  Centauri. 

Two  methods  of  determining  parallax  have  been  applied  in  as- 
tronomy. The  parallax  found  by  one  of  these  methods  is  known  as 
altwlute,  that  by  the  other  as  relative  parallax.  In  determining  the 


DISTANCES  OF  THE  STARS. 


477 


absolute  parallax,  the  observer  finds  the  polar  distance  of  the  star 
as  often  as  possible  through  a  period  of  one  or  more  years  with  a 
meridian  circle,  and  then,  by  a  discussion  of  all  his  observations, 
concludes  what  is  the  magnitude  of  the  oscillation  due  to  parallax. 
The  difficulty  in  applying  this  method  is  that  the  refraction  of  the 
air  and  the  state  of  the  instrument  are  subject  to  changes  arising 
from  varying  temperature,  so  that  the  observations  are  always  un- 
certain by  an  amount  which  is  important  in  such  delicate  work. 

In  determining  the  relative  parallax,  the  astronomer  selects  two 
stars  in  the  same  field  of  view  of  his  telescope,  one  of  which  is 
many  times  more  distant  than  the  other.  It  is  possible  to  judge 
with  a  high  degree  of  probability  which  star  is  the  more  distant, 
from  the  magnitudes  and  proper  motions  of  the  two  objects.  It  is 
assumed  that  a  star  which  is  either  very  bright  or  has  a  large  pro- 
per motion  is  many  times  nearer  to  us  than  the  extremely  faint 
stars  which  may  be  nearly  always  seen  around  it.  The  effect  of 
parallax  will  then  be  to  change  the  apparent  position  of  the  bright 
star  among  the  small  stars  around  it  in  the  course  of  a  year.  This 
change  admits  of  being  measured  with  great  precision  by  the  mi- 
crometer of  the  equatorial,  and  thus  the  relative  parallax  may  be 
determined. 

It  is  true  that  this  relative  parallax  is  really  not  the  absolute  par- 
allax of  either  body,  but  the  difference  of  their  parallaxes.  So  we 
must  necessarily  suppose  that  the  parallax  of  the  smaller  and  more 
distant  object  is  zero.  It  is  by  this  method  of  relative  parallax 
that  the  great  majority  of  determinations  have  been  made. 

The  distances  of  the  stars  are  sometimes  expressed  by 
the  time  required  for  light  to  pass  from  them  to  our  sys- 
tem. The  velocity  of  light  is,  it  will  be  remembered, 
about  300,000  kilometres  per  second,  or  such  as  to  pass 
from  the  sun  to  the  earth  in  8  minutes  18  seconds. 

The  time  required  for  light  to  reach  the  earth  from 
some  of  the  stars,  of  which  the  parallax  has  been  measured, 
is  as  follows  : 


STAK. 

Years. 

STAB. 

Years. 

cr  (Jcntduri            .  .  . 

3-5 

70  Ophiuchi  

19-1 

61  Cygni  

6-7 

4   Ursce  Majoris.  .  .  . 

24-3 

21  185  Lalande 

6-3 

Al'CtUTUS 

25-4 

6  Centauri 

6-9 

7  Draconis  

35-1 

u  Cassiopeium  

9-4 

1830  Groombridge. 

35-9 

84  Oroombridffe 

10-5 

Polaris  

42-4 

21  258  I  alande 

11-9 

3077  Bradley  .   .  . 

4G-1 

17  415  Oeltzen 

13-1 

85  Pectasi     

64-5 

Sirius     .  .       ... 

16-7 

oc.  AuriqcR  

70-1 

17-9 

129-1 

CHAPTER  VII. 

CONSTRUCTION   OF   THE  HEAVENS. 

THE  visible  universe,  as  revealed  to  us  by  the  telescope, 
is  a  collection  of  many  millions  of  stars  and  of  several 
thousand  nebulae.  It  is  sometimes  called  the  stellar  or 
sidereal  system,  and  sometimes,  as  already  remarked,  the 
stellar  universe.  The  most  far-reaching  question  with 
which  astronomy  has  to  deal  is  that  of  the  form  and  mag- 
nitude of  this  system,  and  the  arrangement  of  the  stars 
which  compose  it. 

It  was  once  supposed  that  the  stars  were  arranged  on 
the  same  general  plan  as  the  bodies  of  the  solar  system, 
being  divided  up  into  great  numbers  of  groups  or  clus- 
ters, while  all  the  stars  of  each  group  revolved  in  regular 
orbits  round  the  centre  of  the  group.  All  the  groups  were 
supposed  to  revolve  around  some  great  common  centre, 
which  was  therefore  the  centre  of  the  visible  universe. 

But  there  is  no  proof  that  this  view  is  correct.  The 
only  astronomer  of  the  present  century  who  held  any  such 
doctrine  was  MAEDLER.  He  thought  that  the  centre  of 
motion  of  all  the  stars  was  in  the  Pleiades,  but  no  other 
astronomer  shared  his  views.  We  have  already  seen  that 
a  great  many  stars  are  collected  into  clusters,  but  there  is 
no  evidence  that  the  stars  of  these  clusters  revolve  in 
regular  orbits,  or  that  the  clusters  themselves  have  any 
regular  motion  around  a  common  centre.  Besides,  the 
large  majority  of  stars  visible  with  the  telescope  do  not 
appear  to  be  grouped  into  clusters  at  all. 


STRUCTURE  OF  THE  HEAVENS. 


479 


The  first  astronomer  to  make  a  careful  study  of  the 
arrangement  of  the  stars  with  a  view  to  learn  the  structure 
of  the  heavens  was  Sir  WILLIAM  HEKSCHEL.  He  published 
in  the  Philosophical  Transactions  several  memoirs  on  the 
construction  of  the  heavens  and  the  arrangement  of  the 
stars,  which  have  become  justly  celebrated.  We  s  hall 
therefore  begin  with  an  account  of  HERSCIIEL'S  methods 
and  results. 

HERSCHEL'S  method  of  study  was  founded  on  a  mode  of 
observation  which  he  called  star-gauging.  It  consisted  in 
pointing  a  powerful  telescope  toward  various  parts  of  the 
heavens  and  ascertaining  by  actual  count  how  thick  the 
stars  were  in  each  region.  His  20-foot  reflector  was  pro- 
vided with  such  an  eye-piece  that,  in  looking  into  it,  he 
would  see  a  portion  of  the  heavens  about  15'  in  diameter. 
A  circle  of  this  size  on  the  celestial  sphere  has  about  one 
quarter  the  apparent  surface  of  the  sun,  or  of  the  full 
moon.  On  pointing  the  telescope  in  any  direction,  a 
greater  or  less  number  of  stars  were  nearly  always  visible. 
These  were  counted,  and  the  direction  in  which  the  tele- 
scope pointed  was  noted.  Gauges  of  this  kind  were  made 
in  all  parts  of  the  sky  at  which  he  could  point  his  instru- 
ment, and  the  results  were  tabulated  in  the  order  of  right 
ascension. 

The  following  is  an  extract  from  the  gauges,  and  gives 
the  average  number  of  stars  in  each  field  at  the  points 
noted  in  right  ascension  and  north  polar  distance  : 


N.  P.  D. 

R.  A. 

N.  P.  D. 

R. 

A. 

92°  to  94° 

78°  to  80° 

No.  of  Stars. 

No.  of  Stars. 

h. 
15 

m. 
10 

9-4 

h. 
11 

m. 
6 

3-1 

15 

22 

10-6 

12 

31 

3-4 

15 

47 

10-6 

12 

44 

4-6 

16 

8 

12-1 

12 

49 

3-9 

16 

25 

13-6 

13 

5 

3-8 

16 

37 

18-6 

14 

30 

3-6 

480  ASTRONOMY. 

In  this  small  table,  it  is  plain  that  a  different  law  of 
clustering  or  of  distribution  obtains  in  the  two  regions. 
Such  differences  are  still  more  marked  if  we  compare  the 
extreme  cases  found  by  HERSCHEL,  as  R.  A.  =  19h  41m, 
N.  P.  D.  =  74°  33',  number  of  stars  per  field  ;  588, 
and  E.  A.  =  16h  10m,  K  P.  D.,  113°  4',  number  of 
stars  =  1-1. 

The  number  of  these  stars  in  certain  portions  is  very 
great.  For  example,  in  the  Milky  Way,  near  Orion,  six 
fields  of  view  promiscuously  taken  gave  110,  60,  TO,  90, 
70,  and  74  stars  each,  or  a  mean  of  79  stars  per  field. 
The  most  vacant  space  in  this  neighborhood  gave  63  stars. 
So  that  as  HERSCHEL 's  sweeps  were  two  degrees  wide  in 
declination,  in  one  hour  (15°)  there  would  pass  through 
the  field  of  his  telescope  40,000  or  more  stars.  In  some 
of  the  sweeps  this  number  was  as  great  as  116,000  stars 
in  a  quarter  of  an  hour. 

On  applying  this  telescope  to  the  Milky  Way,  HER- 
SCHEL supposed  at  the  time  that  it  completely  resolved  the 
whole  whitish  appearance  into  small  stars.  This  conclu- 
sion he  subsequently  modified.  He  says  : 

4 '  It  is  very  probable  that  the  great  stratum  called  the  Milky  Way 
is  that  in  which  the  sun  is  placed,  though  perhaps  not  in  the  very 
centre  of  its  thickness. 

"  We  gather  this  from  the  appearance  of  the  Galaxy,  -which 
seems  to  encompass  the  whole  heavens,  as  it  certainly  must  do  if 
the  sun  is  within  it.  For,  suppose  a  number  of  stars  arranged  be- 
tween two  parallel  planes,  indefinitely  extended  every  way,  but  at 
a  given  considerable  distance  from  each  other,  and  calling  this  a 
sidereal  stratum,  an  eye  placed  somewhere  within  it  will  see  all 
the  stars  in  the  direction  of  the  planes  of  the  stratum  projected  into 
a  great  circle,  which  will  appear  lucid  on  account  of  the  accumu- 
lation of  the  stars,  while  the  rest  of  the  heavens,  at  the  sides,  will 
only  seem  to  be  scattered  over  with  constellations,  more  or  less 
crowded,  according  to  the  distance  of  the  planes,  or  number  of 
stars  contained  in  the  thickness  or  sides  of  the  stratum. ' ' 

Thus  in  HERSCHEL' s  figure  an  eye  at  $  within  the  stratum  al 
will  see  the  stars  in  the  direction  of  its  length  a  &,  or  height  c  d, 
with  all  those  in  the  intermediate  situations,  projected  into  the 
lucid  circle  A  C  BD,  while  those  in  the  sides  w#,  nw,  will  be  seen 
scattered  over  the  remaining  part  of  the  heavens  M  VN  W. 


STRUCTURE  OF  THE  HEAVENS. 


481 


u  If  the  eye  were  placed  somewhere  without  the  stratum,  at  no 
very  great  distance,  the  appearance  of  the  stars  within  it  would 
assume  the  form  of  one  of  the  smaller  circles  of  the  sphere,  which 


•    :•.• 

MR"   * 

f;t  w.»  -    • 

7*T    ft.        *  '     ' 


FIG.  135. — HERSCHEL'S  THEORY  OF  THE  STELLAR  SYSTEM. 


would  be  more  or  less  contracted  according  to  the  distance  of  the 
eye  ;  and  if  this  distance  were  exceedingly  increased,  the  whole 
stratum  might  at  last  be  drawn  together  into  a  lucid  spot  of  any 


482  ASTRONOMY. 

shape,  according  to  the  length,  breadth,  and  height  of  the  stra- 
tum. 

' '  Suppose  that  a  smaller  stratum  p  q  should  branch  out  from 
the  former  in  a  certain  direction,  and  that  it  also  is  contained 
between  two  parallel  planes,  so  that  the  eye  is  contained  within 
the  great  stratum  somewhere  before  the  separation,  and  not  far 
from  the  place  where  the  strata  are  still  united.  Then  this  second 
stratum  will  not  be  projected  into  a  bright  circle  like  the  former, 
but  it  will  be  seen  as  a  lucid  branch  proceeding  from  the  first,  and 
returning  into  it  again  at  a  distance  less  than  a  semicircle. 

"In  the  figure  the  stars  in  the  small  stratum  p  q  will  be  pro- 
jected into  a  bright  arc  P  R  R  P,  which,  after  its  separation  from 
the  circle  G  B  D,  unites  with  it  again  at  P. 

"If  the  bounding  surfaces  are  not  parallel  planes,  but  irregularly 
curved  surfaces,  analogous  appearances  must  result." 

The  Milky  Way,  as  we  see  it,  presents  the  aspect  which 
has  been  just  accounted  for,  in  its  general  appearance  of  a 
girdle  around  the  heavens  and  in  its  bifurcation  at  a  cer- 
tain point,  and  HERSCHEL'S  explanation  of  this  appear- 
ance, as  just  given,  has  never  been  seriously  questioned. 
One  doubtful  point  remains :  are  the  stars  in  Fig.  135 
scattered  all  through  the  space  S  —  a  bj}  d  ?  or  are  they 
near  its  bounding  planes,  or  clustered  in  any  way  within 
this  space  so  as  to  produce  the  same  result  to  the  eye  as  if 
uniformly  distributed  ? 

HERSCHEL  assumed  that  they  were  nearly  equably  ar- 
ranged all  through  the  space  in  question.  He  only  exam- 
ined one  other  arrangement — viz.,  that  of  a  ring  of  stars 
surrounding  the  sun,  and  he  pronounced  against  such  an 
arrangement,  for  the  reason  that  there  is  absolutely  noth- 
ing in  the  size  or  brilliancy  of  the  sun  to  cause  us  to  sup- 
pose it  to  be  the  centre  of  such  a  gigantic  system.  No 
reason  except  its  importance  to  us  personally  can  be  alleged 
for  such  a  supposition.  By  the  assumptions  of  Fig.  135, 
each  star  will  have  its  own  appearance  of  a  galaxy  or  milky 
way,  which  will  vary  according  to  the  situation  of  the  star. 

Such  an  explanation  will  account  for  the  general  appear- 
ances of  the  Milky  Way  and  of  the  rest  of  the  sky,  sup- 
posing the  stars  equally  or  nearly  equally  distributed  in 
space.  On  this  supposition,  the  system  must  be  deeper 


STRUCTURE  OF  THE  HEAVENS. 


483 


where  the  stars  appear  more  numerous.  The  same  evi- 
dence can  be  strikingly  presented  in  another  way  so  as  to 
include  the  results  of  the  southern  gauges  of  Sir  JOHN 
HERSCHEL.  The  Galaxy,  or  Milky  Way,  being  nearly  a 
great  circle  of  the  sphere,  we  may  compute  the  position 
of  its  north  or  south  pole ;  and  as  the  position  of  our  own 
polar  points  can  evidently  have  no  relation  to  the  stellar 
universe,  we  express  the  position  of  the  gauges  in  galactic 
polar  distance,  north  or  south.  By  subtracting  these 
polar  distances  from  90°,  we  shall  have  the  distance  of  each 
gauge  from  the  central  plane  of  the  Galaxy  itself,  the  stars 
near  90°  of  polar  distance  being  within  the  Galaxy.  The 
average  number  of  stars  per  field  of  15'  for  each  zone  of 
15°  of  galactic  polar  distance  has  been  tabulated  by  STRUVE 
and  HERSCHEL  as  follows : 


Zones  of  Galactic 
North  Polar 

Average  Number 
of  Stars  per 

Zones  of 
Galactic  South  Polar 

Average  Number 
of  Stars  per 

Distance. 

Field  of  15'. 

Distance. 

Field  of  iy. 

0°   to  15° 

4-32 

0°    to   15° 

6-05 

15°   to  30° 

5-42 

15°    to  30° 

6-62 

30°   to  45° 

8-21 

30°   to  45° 

9-08 

45°   to  60° 

13-61 

45°    to  60° 

13-49 

60°   to  75° 

24-09 

60°   to  75° 

26-29 

75°   to  90° 

53-43 

75°  to  90° 

59-06 

This  table  clearly  shows  that  the  superficial  distribution 
of  stars  from  the  first  to  the  fifteenth  magnitudes  over  the 
apparent  celestial  sphere  is  such  that  the  vast  majority  of 
them  are  in  that  zone  of  30°  wide,  which  includes  the 
Milky  Way.  Other  independent  researches  have  shown 
that  the  fainter  lucid  stars,  considered  alone,  are  also  dis- 
tributed in  greater  number  in  this  zone. 

HERSCHEL  endeavored,  in  his  early  memoirs,  to  find  the  physical 
explanation  of  this  inequality  of  distribution  in  the  theory  of  the 
universe  exemplified  in  Fig.  136,  which  was  based  on  the  funda- 
mental assumption  that,  oil  the  whole,  the  stars  were  nearly  equably 
distributed  in  space. 


484  ASTRONOMY. 

If  they  were  so  distributed,  then  the  number  of  stars  visible  in 
any  gauge  would  show  the  thickness  of  the  stellar  system  in  the 
direction  in  which  the  telescope  was  pointed.  At  each  pointing, 
the  field  of  view  of  the  instrument  includes  all  the  visible  stars  sit- 
uated within  a  cone,  having  its  vertex  at  the  observer's  eye,  and  its 
base  at  the  very  limits  of  the  system,  the  angle  of  the  cone  (at  the 
eye)  being  15'  4".  Then  the  cubes  of  the  perpendiculars  let  fall 
from  the  eye  on  the  plane  of  the  bases  of  the  various  visual  cones 
are  proportional  to  the  solid  contents  of  the  cones  themselves,  or,  as 
the  stars  are  supposed  equally  scattered  within  all  the  cones,  the 
cube  roots  of  the  numbers  of  stars  in  each  of  the  fields  express  the 
relative  lengths  of  the  perpendiculars.  A  section  of  the  sidereal  sys- 
tem along  any  great  circle  can  thus  be  constructed  as  in  the  figure, 
which  is  copied  from  HERSCHEL. 

The  solar  system  is  supposed  to  be  at  the  dot  within  the  mass  of 
stars.  From  this  point  lines  are  drawn  along  the  directions  in 
which  the  gauging  telescope  was  pointed.  On  these  lines  are  laid 
off  lengths  proportional  to  the  cube  roots  of  the  number  of  stars  in 
each  gauge. 


FlG.    136. — ARRANGEMENT    OP  THE  STARS  ON    THE    HYPOTHESIS  OP 
EQUABLE  DISTRIBUTION. 

The  irregular  line  joining  the  terminal  points  is  approximately 
the  bounding  curve  of  the  stellar  system  in  the  great  circle  chosen. 
Within  this  line  the  space  is  nearly  uniformly  filled  with  stars. 
Without  it  is  empty  space.  A  similar  section  can  be  constructed  in 
any  other  great  circle,  and  a  combination  of  all  such  would  give  a 
representation  of  the  shape  of  our  stellar  system.  The  more  numer- 
ous and  careful  the  observations,  the  more  elaborate  the  represen- 
tation, and  the  863  gauges  of  HERSCHEL  are  sufficient  to  mark  out 
with  great  precision  the  main  features  of  the  Milky  Way,  and  even 
to  indicate  some  of  its  chief  irregularities.  This  figure  may  be 
compared  with  Fig.  135. 

On  the  fundamental  assumption  of  HERSCHEL  (equable  distribu- 
tion), no  other  conclusions  can  be  drawn  from  his  statistics  but 
that  drawn  by  him. 

This  assumption  he  subsequently  modified  in  some  degree,  and 
was  led  to  regard  his  gauges  as  indicating  not  so  much  the  depth 
of  the  system  in  any  direction  as  the  clustering  power  or  tendency 
of  the  stars  in  those  special  regions.  It  is  clear  that  if  in  any 


STRUCTURE  OF  THE  HEAVENS.  485 

given  part  of  the  sky,  where,  on  the  average,  there  are  10  stars 
(say)  to  a  field,  we  should  find  a  certain  small  portion  of  100  or 
more  to  a  field,  then,  on  HERSCHEL'S  first  hypothesis,  rigorously  in- 
terpreted, it  would  be  necessary  to  suppose  a  spike-shaped  protu- 
berance directed  from  the  earth  in  order  to  explain  the  increased 
number  of  stars.  If  many  such  places  could  be  found,  then  the 
probability  is  great  that  this  explanation  is  wrong.  We  should 
more  rationally  suppose  some  real  inequality  of  star  distribution 
here.  It  is,  in  fact,  in  just  such  details  that  the  system  of  HER- 
SCHEL  breaks  down,  and  the  careful  examination  which  his  system 
has  received  leads  to  the  belief  that  it  must  be  greatly  modified  to 
cover  all  the  known  facts,  while  it  undoubtedly  has,  in  the  main,  a 
strong  basis. 

The  stars  are  certainly  not  uniformly  distributed,  and  any  gen- 
eral theory  of  the  sidereal  system  must  take  into  account  the  varied 
tendency  to  aggregation  in  various  parts  of  the  sky. 

The  curious  convolutions  of  the  Milky  Way,  observed  at  various 
parts  of  its  course,  seem  inconsistent  with  the  idea  of  very  great 
depth  of  this  stratum,  and  Mr.  PROCTOR  has  pointed  out  that  the 
circular  forms  of  the  two  "  coal-sacks"  of  the  Southern  Milky  Way 
indicate  that  they  are  really  globular,  instead  of  being  cylindric 
tunnels  of  great  length,  looking  into  space,  with  their  axes  directed 
toward  the  earth.  If  they  are  globular,  then  the  depth  of  the 
Milky  Way  in  their  neighborhood  cannot  be  greatly  different  from 
their  diameters,  which  would  indicate  a  much  smaller  depth  than 
that  assigned  by  HERSCHEL. 

In  1817,  HERSCHEL  published  an  important  memoir  on  the  same 
subject,  in  which  his  first  method  was  largely  modified,  though 
not  abandoned  entirely.  Its  fundamental  principle  was  stated  by 
him  as  follows  : 

"  It  is  evident  that  we  cannot  mean  to  affirm  that  the  stars  of  the 
fifth,  sixth,  and  seventh  magnitudes  are  really  smaller  than  those 
of  the  first,  second,  or  third,  and  that  we  must  ascribe  the  cause 
of  the  difference  in  the  apparent  magnitudes  of  the  stars  to  a  differ- 
ence in  their  relative  distances  from  us.  On  account  of  the  great 
number  of  stars  in  each  class,  we  must  also  allow  that  the  stars  of 
each  succeeding  magnitude,  beginning  with  the  first,  are,  one  with 
another,  further  from  us  than  those  of  the  magnitude  immediately 
preceding.  The  relative  magnitudes  give  only  relative  distances, 
and  can  afford  no  information  as  to  the  real  distances  at  which  the 
stars  are  placed. 

"  A  standard  of  reference  for  the  arrangement  of  the  stars  may 
be  had  by  comparing  their  distribution  to  a  certain  properly  mod- 
ified equality  of  scattering.  The  equality  which  I  propose  does  not 
require  that  the  stars  should  be  at  equal  distances  from  each  other, 
nor  is  it  necessary  that  all  those  of  the  same  nominal  magnitude 
should  be  equally  distant  from  us." 

It  consists  of  allotting  a  certain  equal  portion  of  space  to  every 
star,  so  that,  on  the  whole,  each  equal  portion  of  space  writhin  the 
stellar  system  contains  an  equal  number  of  stars. 


486 


ASTRONOMY. 


The  space  about  each  star  can  be  considered  spherical.  Sup- 
pose such  a  sphere  to  surround  our  own  sun,  its  radius  will  not 

differ  greatly  from  the 
distance  of  the  nearest 
fixed  star,  and  this  is 
taken  as  the  unit  of 
distance. 

Suppose  a  series  of 
larger  spheres,  all 
drawn  around  our  sun 
as  a  centre,  and  having 
the  radii  3,  5,  7,  9, 
etc.  The  contents  of 
the  spheres  being  as 
the  cubes  of  their 
diameters,  the  first 
sphere  will  have  3x3 
x  3  =  27  times  the 
volume  of  the  unit 
sphere,  and  will  there- 
fore be  large  enough 
to  contain  27  stars  ; 
the  second  will  have 
125  times  the  volume, 
and  will  therefore  con- 
tain 125  stars,  and  so 
with  the  successive 
spheres.  The  figure 
shows  a  section  of 
portions  of  these 
spheres  up  to  that 
with  radius  11.  Above 
the  centre  are  given 
the  various  orders  of 
stars  which  are  situ- 
ated between  the  sev- 
eral spheres,  while 
in  the  correspondin  ' 
spaces  below  the  cen- 
tre are  given  the  num- 
ber of  stars  which  the  region  is  large  enough  to  contain  ;  for  in- 
stance, the  sphere  of  radius  7  has  room  for  343  stars,  but  of  this 
space  125  parts  belong  to  the  spheres  inside  of  it  :  there  is,  there- 
fore, room  for  218  stars  between  the  spheres  of  radii  5  and  7. 

HERSCHEL  designates  the  several  distances  of  these  layers  of 
stars  as  orders  ;  the  stars  between  spheres  1  and  3  are  of  the  first 
order  of  distance,  those  between  3  and  5  of  the  second  order,  and 
so  on.  Comparing  the  room  for  stars  between  the  several  spheres 
with  the  number  of  stars  of  the  several  magnitudes,  he  found  the 
result  to  be  as  follows  : 


PlG.  137. — ORDEKS   OP  DISTANCE   OF  STAKS. 


STRUCTURE  OF  THE  HEAVENS. 


487 


Order  of  Distance. 

Number  of  Stars 
there  is  Room  for. 

Magnitude. 

Number  of  Stars 
of  that  Magnitude. 

•i 

26 

1 

17 

2 

98 

2 

57 

3  

218 

3 

206 

4 

386 

4 

454 

5 

602 

5 

1  161 

c  

866 

0 

6  103 

7  

1,178 

6  146 

8 

1  538 

The  result  of  this  comparison  is,  that,  if  the  order  of  magnitudes 
could  indicate  the  distance  of  the  stars,  it  would  denote  at  first  a 
gradual  and  afterward  a  very  abrupt  condensation  of  them. 

If,  on  the  ordinary  scale  of  magnitudes,  we  assume  the  brightness 
of  any  star  to  be  inversely  proportional  to  the  square  of  its  dis- 
tance, it  leads  to  a  scale  of  distance  different  from  that  adopted  by 
HERSCHEL,  so  that  a  sixth- magnitude  star  on  the  common  scale 
would  be  about  of  the  eighth  order  of  distance  according  to  this 
scheme — that  is,  we  must  remove  a  star  of  the  first  magnitude  to 
eight  times  its  actual  distance  to  make  it  shine  like  a  star  of  the 
sixth  magnitude. 

On  the  scheme  here  laid  down,  HERSCHEL  subsequently  assigned 
the  order  of  distance  of  various  objects,  mostly  star-clusters,  and 
his  estimates  of  these  distances  are  still  quoted.  They  rest  on  the 
fundamental  hypothesis  which  has  been  explained,  and  the  error 
in  the  assumption  of  equal  brilliancy  for  all  stars,  affects  these  esti- 
mates. It  is  perhaps  most  probable  that  the  hypothesis,  of  equal 
brilliancy  for  all  stars  is  still  more  erroneous  than  the  hypothesis 
of  equal  distribution,  and  it  may  well  be  that  there  is  a  very  large 
range  indeed  in  the  actual  dimensions  and  in  the  intrinsic  brilliancy 
of  stars  at  the  same  order  of  distance  from  us,  so  that  the  tenth- 
magnitude  stars,  for  example,  may  be  scattered  throughout  the 
spheres,  which  HERSCHEL  would  assign  to  the  seventh,  eighth, 
ninth,  tenth,  eleventh,  twelfth,  and  thirteenth  magnitudes. 

Since  the  time  of  HERSCHEL,  one  of  the  most  eminent  of  the  as- 
tronomers who  have  investigated  this  subject  is  STRUVE  the  elder, 
formerly  director  of  the  Pulkowa  Observatory.  His  researches 
were  founded  mainly  on  the  numbers  of  stars  of  the  several  magni- 
tudes found  by  BESSEL  in  a  zone  thirty  degrees  wide  extending  all 
around  the  heavens,  15°  on  each  side  of  the  equator.  With  these 
he  combined  the  gauges  of  Sir  WILLIAM  HERSCHEL.  The  hypothesis 
on  which  he  based  his  theory  was  similar  to  that  employed  by 
HERSCHEL  in  his  later  researches,  in  so  far  that  he  supposed  the 
magnitude  of  the  stars  to  furnish,  on  the  average,  a  measure  of 
their  relative  distances.  Supposing,  after  HERSCHEL,  a  number  of 
concentric  spheres  to  be  drawn  around  the  sun  as  a  centre,  the  suc- 
cessive spaces  between  which  corresponded  to  stars  of  the  several 


488 


ASTRONOMY. 


magnitudes,  he  found  that  the  further  out  he  went,  the  more  the 
stars  were  condensed  in  and  near  the  Milky  Way.  This  conclusion 
may  be  drawn  at  once  from  the  fact  we  have  already  mentioned, 
that  the  smaller  the  stars,  the  more  they  are  condensed  in  the  re- 
gion of  the  Galaxy.  STRUVE  found  that  if  we  take  only  the  stars 
plainly  visible  to  the  naked  eye — that  is,  those  down  to  the  fifth 
magnitude — they  are  no  thicker  in  the  Milky  Way  than  in  other 
parts  of  the  heavens.  But  those  of  the  sixth  magnitude  are  a 
little  thicker  in  that  region,  those  of  the  seventh  yet  thicker,  and 
so  on,  the  inequality  of  distribution  becoming  constantly  greater  as 
the  telescopic  power  is  increased. 

From  all  this,  STRUVE  concluded  that  the  stellar  system  might 
be  considered  as  composed  of  lasers  of  stars  of  various  densities,  all 
parallel  to  the  plane  of  the  Milky  Way.  The  stars  are  thickest  in  and 
near  the  central  layer,  which  he  conceives  to  be  spread  out  as  a  wide, 
thin  sheet  of  stars.  Our  sun  is  situated  near  the  middle  of  this 
layer.  As  we  pass  out  of  this  layer,  on  either  side  we  find  the 
stars  constantly  growing  thinner  and  thinner,  but  we  do  not  reach 
any  distinct  boundary.  As,  if  we  could  rise  in  the  atmosphere,  we 
should  find  the  air  constantly  growing  thinner,  but  at  so  gradual  a 
rate  of  progress  that  we  could  hardly  say  where  it  terminated  ;  so, 
on  STRUVE' s  view,  would  it  be  with  the  stellar  system,  if  we  could 
mount  up  in  a  direction  perpendicular  to  the  Milky  Way.  STRUVE 
gives  the  following  table  of  the  thickness  of  the  stars  on  each  side 
of  the  principal  plane,  the  unit  of  distance  being  that  of  the  ex- 
treme distance  to  which  HERSCHEL'S  telescope  could  penetrate  : 


Distance  from  Principal  Plane. 

Density. 

Mean  Distance 
between  Neighbor- 
ing Stars. 

In  the  prin 
0-05  from 
0-10 
0-20 
0-30 
0-40 
0-50 
0-60 
0-70 
0-80 
0-866 

cipal  pla 
principal 

ne  

1-0000 
0-48568 
0-33288 
0-23895 
0-17980 
0-13021 
0-08646 
0-05510 
0-03079 
0-01414 
0-00532 

1-000 
1-272 
1-458 
1-611 
1-772 
1-973 
2-261 
2-628 
3-190 
4-131 
5-729 

plane   



This  condensation  of  the  stars  near  the  central  plane  and  the 
gradual  thinning-out  on  each  side  of  it  are  only  designed  to  be  the 
expression  of  the  general  or  average  distribution  of  those  bodies. 
The  probability  is  that  even  in  the  central  plane  the  stars  are  many 
times  as  thick  in  some  regions  as  in  others,  and  that,  as  we  leave  the 
plane,  the  thinning-out  would  be  found  to  proceed  at  very  different 
rates  in  different  regions.  That  there  may  be  a  gradual  thinning-out 


STRUCTURE  OF  THE  HEAVENS.  489 

cannot  be  denied  ;  but  STRUVE'S  attempt  to  form  a  table  of  it  is  open 
to  the  serious  objection  that,  like  HERSCHEL,  he  supposed  the  differ- 
ences between  the  magnitudes  of  the  stars  to  arise  entirely  from 
their  different  distances  from  us.  Although  where  the  scattering 
of  the  stars  is  nearly  uniform,  this  supposition  may  not  lead  us  into 
serious  error,  the  case  will  be  entirely  different  where  we  have  to 
deal  with  irregular  masses  of  stars,  and  especially  where  our  tele- 
scopes penetrate  to  the  boundary  of  the  stellar  system.  In  the 
latter  case  we  cannot  possibly  distinguish  between  small  stars  lying 
within  the  boundary  and  larger  ones  scattered  outside  of  it,  and 
STRUVE'S  gradual  thinning-out  of  the  stars  may  be  entirely  ac- 
counted for  by  great  diversities  in  the  absolute  brightness  of  the 
stars. 

Distribution  of  Stars.  —  The  brightness  B  of  any  star,  as  seen 
from  the  earth,  depends  upon  its  surface  8,  the  intensity  of  its  light 
per  unit  of  surface,  i,  and  its  distance  D,  so  that  its  brightness  can  be 
expressed  thus  : 


_,       S'  x  i' 
for  another  star  :  B   =  —j-  > 


B  S  •  i 

and  -     =     - 


Now  this  ratio  of  the  brightness  B  -J-  B'  is  the  only  fact  we  usually 
know  with  regard  to  any  two  stars.  D  has  been  determined  for 
only  a  few  stars,  and  for  these  it  varies  between  200,000  and  2,000,000 
times  the  major  axis  of  the  earth's  orbit.  S  and  i  are  not  known  for 
any  star.  There  is,  however,  a  probability  that  i  does  not  vary  greatly 
from  star  to  star,  as  the  great  majority  of  stars  are  white  in  color  (only 
some  700  red  stars,  for  instance,  are  known  out  of  the  300,000  which 
have  been  carefully  examined).  Among  476  double  stars  of  STRUVE'S 
list  295  were  white,  63  being  bluish,  only  one  fourth,  or  118,  being 
yellow  or  red. 

If  B  is  of  the  nth  mag.  its  light  in  terms  of  a  first  magnitude  star 
is  6n  -  l  where  6  =  0-397,  and  if  B'  is  of  the  rath  mag.,  its  light  is 
$m—  i?  both  expressed  in  terms  of  the  light  of  a  first  magnitude  star  as 
unity  (d°  =  1). 

Therefore  we  may  put  B  —  6n  —  ',  B'  =.  <J"l~  l,  and  we  have 


d»-i  :    $'  -i-  IP 

In  this  general  expression  we  seek  the  ratio  -j^-,    and   we   have   it 

expressed  in  terms  of  four  unknown  quantities.     We  must  therefore 
make  some  supposition  in  regard  to  these. 

I.  If  all  stars  are  of  equal  intrinsic  brilliancy  and  of  equal  size,  then 

Si,  S'  i',  and  6n  ~m  =  a  constant  =  — ™-, 


490  ASTRONOMY. 

whence  the  relative  distance  of  any  two  stars  would  be  known  on  this 
hypothesis. 

II.  Or,  suppose  the  stars  to  be  urfiformly  distributed  in  space,  or  the 
star-density  to  be  equal  in  all  directions.  From  this  we  can  also 
obtain  some  notions  of  the  relative  distances  of  stars. 

Call  Di,  D2,  D3 Dn  the  average  distances  of  stars  of  the 

1,  2,  3, ftth  magnitudes. 

If  K  stars  are  situated  within  the  sphere  of  radius  1,  then  the  num- 
ber of  stars  (Qn),  situated  within  the  sphere  of  radius  Dn,  is 

Qn  =  K  •  (Dn)3t 

since  the  cubic  contents  of  spheres  are  as  the  cubes  of  their  radii. 
Also 

Qn  __  ,  =  K  (Dn  -  ,)3, 
whence 


If  we  knew  Qn  and  Qn  —  i,  the  number  of  stars  contained  in  the 
spheres  of  radii  Dn  and  Dn  -  i,  then  the  ratio  of  Dn  and  Dn  _  j  would 
be  known.  We  cannot  know  QH,  Qn  -  \,  etc.,  directly,  but  we  may 
suppose  these  quantities  to  be  proportional  to  the  numbers  of  stars  of 
the  ntli  and  (n  —  l)th  magnitudes  found  in  an  enumeration  of  all  the 
stars  in  the  heavens  of  tluese  magnitudes,  or,  failing  in  these  data,  we 
may  confine  this  enumeration  to  the  northern  hemisphere,  where 
LITTROW  has  counted  the  number  of  stars  of  each  class  in  ARGELLAN- 
DEII'S  Dwrchmusterung.  As  we  have  seen  (p.  436) 

Q,  =  19,699  and  Q6  =  77,794, 
whence 

~ir~ =  ^~ir  =  1'58> 

and  this  would  lead  us  to  infer  that  the  stars  of  the  8th  magnitude 
were  distributed  inside  of  a  sphere  whose  radius  was  about  1-6  times 
that  of  the  corresponding  sphere  for  the  7th  magnitude  stars  provided 
that,  1st,  the  stars  in  general  are  equally  or  about  equally  distributed, 

and,  2d,  that  on  the  whole  the   stars  of  the  8 n  magnitudes  are 

further  away  from  us  than  those  of  the  7 (n  —  1)  magnitudes. 

We  may  have  a  kind  of  test  of  the  truth  of  this  hypothesis,  and  of 
the  first  employed,  as  follows,  we  had  : 


Also  from  the  first  hypothesis  the  brightness  Bn  of  a  star  of  the  ntli 
magnitude  in  terms  of  a  first  magnitude  star  =  1  was 


If  here,  again,  we  suppose   the  distance  of  a  first  magnitude  star  to 
be  =  1  and  of  an  nth  magnitude  star  Dn,  then 


STRUCTURE  OF  THE  HEAVENS. 
*.:        * 


491 


or 

Also 

whence 


D 


Comparing  the  expression  for  —  -- — ,  in  the  two  cases,  we  have 

JDn  —  1 


V- 


__ 

Qn-l 


-""•=          ^ 


If  the  value  of  6  in  this  last  expression  comes  near  to  the  value  which 
has  been  deduced  for  it  from  direct  photometric  measures  of  the 
relative  intensity  of  various  classes  of  stars,  viz.,  6  =  0-40,  then  this 
will  be  so  far  an  argument  to  show  that  a  certain  amount  of  credence 
may  be  given  to  both  hypotheses  I.  and  II.  Taking  the  values  of 
Qi  and  Q6,  we  have 


3  (7,  „)  - 


19,699 


T-        =  0-40. 


From  the  values  of  Q6  and  Q7,  there  results  6(6, 7)  =  0-45.  These, 
then,  agree  tolerably  well  with  the  independent  photometric  values 
for  6,  and  show  that  the  equation 


\  VA~  ) 


gives  the  average  distance  of  the  stars  of  the  nth  magnitude  with  a 
certain  approach  to  accuracy.  For  the  stars  from  1st  to  8th  magni- 
tude these  distances  are  : 

1  to  1-9  magnitude 1-00 


2  to  2-9 

3  to  3-9 

4  to  4-9 

5  to  5-9 

6  to  6-9 

7  to  7-9 

8  to  8-9 


1-54 

2-36 

3-64 

5-59 

8-61 

13-23 

20-35 


This  presentation  of  the   subject  is  essentially  that  of  Prof,   HUGO 
GYLDEN. 


CHAPTER    VIII. 

COSMOGONY. 

A  THEOKY  of  the  operations  by  which  the  universe  re- 
ceived its  present  form  and  arrangement  is  called  Cosmog- 
ony. This  subject  does  not  treat  of  the  origin  of  matter, 
but  only  with  its  transformations. 

Three  systems  of  Cosmogony  have  prevailed  among 
thinking  men  at  different  times. 

(1.)  That  the  universe  had  no  origin,  but  existed  from 
eternity  in  the  form  in  which  we  now  see  it. 

(2.)  That  it  was  created  in  its  present  shape  in  a 
moment,  out  of  nothing. 

(3.)  That  it  came  into  its  present  form  through  an  ar- 
rangement of  materials  which  were  before  i '  without  form 
and  void." 

The  last  seems  to  be  the  idea  which  has  most  prevailed 
among  thinking  men,  and  it  receives  many  striking  con- 
firmations from  the  scientific  discoveries  of  modern  times. 
The  latter  seem  to  show  beyond  all  reasonable  doubt  that 
the  universe  could  not  always  have  existed  in  its  present 
form  and  under  its  present  conditions  ;  that  there  was  a  time 
when  the  materials  composing  it  were  masses  of  glowing 
vapor,  and  that  there  will  be  a  time  when  the  present  state 
of  things  will  cease.  The  explanation  of  the  processes 
through  which  this  occurs  is  sometimes  called  the  nebular 
hypothesis.  It  was  first  propounded  by  the  philosophers 
SWEDENBORG,  KANT,  and  LAPLACE,  and  although  since 
greatly  modified  in  detail,  the  views  of  these  men  have  in 
the  main  been  retained  until  the  present  time. 


COSMOGONY.  493 

We  shall  begin  its  consideration  by  a  statement  of  the 
various  facts  which  appear  to  show  that  the  earth  and 
planets,  as  well  as  the  sun,  were  once  a  fiery  mass. 

The  first  of  these  facts  is  the  gradual  but  uniform  in- 
crease of  temperature  as  we  descend  into  the  interior  of 
the  earth.  Wherever  mines  have  been  dug  or  wells  Bunk 
to  a  great  depth,  it  is  found  that  the  temperature  increases 
as  we  go  downward  at  the  rate  of  about  one  degree  centi- 
grade to  every  30  metres,  or  one  degree  Fahrenheit  to 
every  50  feet.  The  rate  differs  in  different  places,  but  the 
general  average  is  near  this.  The  conclusion  which  we 
draw  from  this  may  not  at  first  sight  be  obvious,  because 
it  may  seem  that  the  earth  might  always  have  shown  this 
same  increase  of  temperature.  But  there  are  several  re- 
sults which  a  little  thought  will  make  clear,  although  their 
complete  establishment  requires  the  use  of  the  higher 
mathematics. 

The  first  result  is  that  the  increase  of  temperature  can- 
not be  merely  superficial,  but  must  extend  to  a  great 
depth,  probably  even  to  the  centre  of  the  earth.  If  it  did 
not  so  extend,  the  heat  would  have  all  been  lost  loner  ages 

'  O        O 

ago  by  conduction  to  the  interior  and  by  radiation  from 
the  surface.  It  is  certain  that  the  earth  has  not  received 
any  great  supply  of  heat  from  outside  since  the  earliest 
geological  ages,  because  such  an  accession  of  heat  at  the 
earth's  surface  would  have  destroyed  all  life,  and  even 
melted  all  the  rocks.  Therefore,  whatever  heat  there  is 
in  the  interior  of  the  earth  must  have  been  there  from  be- 
fore the  commencement  of  life  on  the  globe,  and  remained 
through  all  geological  ages. 

The  interior  of  the  earth  being  hotter  than  its  surface, 
and  hotter  than  the  space  around  it,  must  be  losing  heat. 
We  know  by  the  most  familiar  observation  that  if  any  ob- 
ject is  hot  inside,  the  heat  will  work  its  way  through  to  the 
surface  by  the  process  of  conduction.  Therefore,  since  the 
earth  is  a  great  deal  hotter  at  the  depth  of  30  metres  than 
it  is  at  the  surface,  heat  must  be  continually  coming  to  the 


494  ASTRONOMY. 

surface.  On  reaching  the  surface,  it  must  he  radiated  off 
into  space,  else  the  surface  WQuld  have  long  ago  become 
as  hot  as  the  interior.  Moreover,  this  loss  of  heat  must 
have  been  going  on  since  the  beginning,  or,  at  least,  since 
a  time  when  the  surface  was  as  hot  as  the  interior.  Thus,  if 
we  reckon  backward  in  time,  we  find  that  there  must  have 
been  more  and  more  heat  in  the  earth  the  further  back 
we  go,  so  that  we  must  finally  reach  back  to  a  time  when 
it  was  so  hot  as  to  be  molten,  and  then  again  to  a  time 
when  it  was  so  hot  as  to  be  a  mass  of  fiery  vapor. 

The  second  fact  is  that  we  find  the  sun  to  be  cooling  off 
like  the  earth,  only  at  an  incomparably  more  rapid  rate. 
The  sun  is  constantly  radiating  heat  into  space,  and,  so  far 
as  we  can  ascertain,  receiving  none  back  again.  A  small 
portion  of  this  heat  reaches  the  earth,  and  on  this  portion 
depends  the  existence  of  life  and  motion  on  the  earth's  sur- 
face. The  quantity  of  heat  which  strikes  the  earth  is  only 
about  -g-p-g- 00*00000  °^  *na*  which  the  sun  radiates.  This 
fraction  expresses  the  ratio  of  the  apparent  surface  of  the 
earth,  as  seen  from  the  sun,  to  that  of  the  whole  celestial 
sphere. 

Since  the  sun  is  losing  heat  at  this  rate,  it  must  have  had 
more  heat  yesterday  than  it  has  to-day  ;  more  two  days  ago 
than  it  had  yesterday,  and  so  on.  Thus  calculating  back- 
ward, we  find  that  the  further  we  go  back  into  time  the 
hotter  the  sun  must  have  been.  Since  we  know  that  heat 
expands  all  bodies,  it  follows  that  the  sun  must  have  been 
larger  in  past  ages  than  it  is  now,  and  we  can  trace  back 
this  increase  in  size  without  limit.  Thus  we  are  led  to  the 
conclusion  that  there  must  have  been  a  time  when  the  sun 
filled  up  the  space  now  occupied  by  the  planets,  and  must 
have  been  a  very  rare  mass  of  glowing  vapor.  The  plan- 
ets could  not  then  have  existed  separately,  but  must  have 
formed  a  part  of  this  mass  of  vapor.  The  latter  was  there- 
fore the  material  out  of  which  the  solar  system  was 
formed. 

The  same  process  may  be  continued  into  the  future. 


COSMOGONY.  495 

Since  the  sun  by  its  radiation  is  constantly  losing  heat,  it 
must  grow  cooler  and  cooler  as  ages  advance,  and  must 
finally  radiate  so  little  heat  that  life  and  motion  can  no 
longer  exist  on  our  globe. 

The  third  fact  is  that  the  revolutions  of  all  the  planets 
around  the  sun  take  place  in  the  same  direction  and  in 
nearly  the  same  plane.  We  have  here  a  similarity  amongst 
the  different  bodies  of  the  solar  system,  which  must  have 
had  an  adequate  cause,  and  the  only  cause  which  has  ever 
been  assigned  is  found  in  the  nebular  hypothesis.  This 
hypothesis  supposes  that  the  sun  and  planets  were  once 
a  great  mass  of  vapor,  as  large  as  the  present  solar  system, 
revolving  on  its  axis  in  the  same  plane  in  which  the 
planets  now  revolve. 

The  fourth  fact  is  seen  in  the  existence  of  nebulae.  We 
have  already  stated  that  the  spectroscope  shows  these  bodies 
to  be  masses  of  glowing  vapor.  We  thus  actually  see  mat- 
ter in  the  celestial  spaces  under  the  very  form  in  which 
the  nebular  hypothesis  supposes  the  matter  of  our  solar 
system  to  have  once  existed.  Since  these  masses  of  vapor 
are  so  hot  as  to  radiate  light  and  heat  through  the  immense 
distance  which  separates  us  from  them,  they  must  be  grad- 
ually cooling  off.  This  cooling  must  at  length  reach  a 
point  when  they  will  cease  to  be  vaporous  and  condense 
into  objects  like  stars  and  planets.  We  know  that  every 
star  in  the  heavens  radiates  heat  as  our  sun  does.  In  the 
case  of  the  brighter  stars  the  heat  radiated  has  been  made 
sensible  in  the  foci  of  our  telescopes  by  means  of  the  thermo- 
multiplier.  The  general  relation  which  we  know  to  ex- 
ist between  light  and  radiated  heat  shows  that  all  the  stars 
must,  like  the  sun,  be  radiating  heat  into  space. 

A  fifth  fact  is  afforded  by  the  physical  constitution  of 
the  planets  Jupiter  and  Saturn.  The  telescopic  examina- 
tion of  these  planets  shows  that  changes  on  their  surfaces 
are  constantly  going  on  with  a  rapidity  and  violence  to 
which  nothing  on  the  surface  of  our  earth  can  compare. 
Such  operations  can  be  kept  up  only  through  the  agency  of 


496  ASTRONOMY. 

heat  or  some  equivalent  form  of  energy.  But  at  the  dis- 
tance of  Jupiter  and  Saturn  the  rajs  of  the  sun  are  entirely 
insufficient  to  produce  changes  so  violent.  We  are  there- 
fore led  to  infer  that  Jupiter  and  Saturn  must  be  hot 
bodies,  and  must  therefore  be  cooling  off  like  the  sun, 
stars  and  earth. 

We  are  thus  led  to  the  general  conclusion  that,  so  far 
as  our  knowledge  extends,  nearly  all  the  bodies  of  the 
universe  are  hot,  and  are  cooling  off  by  radiating  their 
heat  into  space.  Before  the  discovery  of  the  "  conserva- 
tion of  energy,"  it  was  not  known  that  this  radiation  in- 
volved the  waste  of  a  something  which  is  necessarily  limited 
in  supply.  But  it  is  now  known  that  heat,  motion,  and 
other  forms  of  force  are  to  a  certain  extent  convertible  into 
each  other,  and  admit  of  being  expressed  as  quantities  of 
a  general  something  which  is  called  energy.  We  may  de- 
fine the  unit  of  energy  in  two  or  more  ways  :  as  the  quan- 
tity which  is  required  to  raise  a  certain  weight  through  a 
certain  height  at  the  surface  of  the  earth,  or  to  heat  a  given 
quantity  of  water  to  a  certain  temperature.  However 
we  express  it,  we  know  by  the  laws  of  matter  that  a  given 
mass  of  matter  can  contain  only  a  certain  definite  number 
of  units  of  energy.  When  a  mass  of  matter  either  gives 
off  heat,  or  causes  motion  in  other  bodies,  we  know  that 
its  energy  is  being  expended.  Since  the  total  quantity  of 
energy  which  it  contains  is  finite,  the  process  of  radiating 
heat  must  at  length  come  to  an  end. 

It  is  sometimes  supposed  that  this  cooling  off  may  be 
merely  a  temporary  process,  and  that  in  time  something 
may  happen  by  which  all  the  bodies  of  the  universe  will 
receive  back  again  the  heat  which  they  have  lost.  This  is 
founded  upon  the  general  idea  of  a  compensating  process  in 
nature.  As  a  special  example  of  its  application,  some  have 
supposed  that  the  planets  may  ultimately  fall  into  the  sun, 
and  thus  generate  so  much  heat  as  to  reduce  the  sun  once 
more  to  vapor.  All  these  theories  are  in  direct  opposition 
to  the  well-established  laws  of  heat,  and  can  be  justified 


COSMOGONY.  497 

only  by  some  generalization  which  shall  be  far  wider  than 
any  that  science  has  yet  reached.  Until  we  have  such  a 
generalization,  every  such  theory  founded  upon  or  consist- 
ent with  the  laws  of  nature  is  a  necessary  failure.  All  the 
heat  that  could  be  generated  by  a  fall  of  all  the  planets  into 
the  sun  would  not  produce  any  change  in  its  constitution, 
and  would  only  last  a  few  years.  The  idea  that  the  heat 
radiated  by  the  sun  and  stars  may  in  some  way  be  collected 
and  returned  to  them  by  the  mere  operation  of  natural  laws 
is  equally  untenable.  It  is  a  fundamental  principle  of  the 
laws  of  heat  that  the  latter  can  never  pass  from  a  cooler 
to  a  warmer  body,  and  that  a  body  can  never  grow  warm 
or  acquire  heat  in  a  space  that  is  cooler  than  the  body  is 
itself.  All  differences  of  temperature  tend  to  equalize 
themselves,  and  the  only  state  of  things  to  which  the  uni- 
verse can  tend,  under  its  present  laws,  is  one  in  which  all 
space  and  all  the  bodies  contained  in  space  are  at  a  uniform 
temperature,  and  then  all  motion  and  change  of  tempera- 
ture, and  hence  the  conditions  of  vitality,  must  cease.  And 
then  all  such  life  as  ours  must  cease  also  unless  sustained 
by  entirely  new  methods. 

The  general  result  drawn  from  all  these  laws  and  facts 
is,  that  there  was  once  a  time  when  all  the  bodies  of  the 
universe  formed  either  a  single  mass  or  a  number  of  masses 
of  fiery  vapor,  having  slight  motions  in  various  parts,  and 
different  degrees  of  density  in  different  regions.  A  grad- 
ual condensation  around  the  centres  of  greatest  density  then 
went  on  in  consequence  of  the  cooling  and  the  mutual  at- 
traction of  the  parts,  and  thus  arose  a  great  number  of 
nebulous  masses.  One  of  these  masses  formed  the  ma- 
terial out  of  which  the  sun  and  planets  are  supposed  to 
have  been  formed.  It  was  probably  at  first  nearly  glob- 
ular, of  nearly  equal  density  throughout,  and  endowed 
with  a  very  slow  rotation  in  the  direction  in  which  the 
planets  now  move.  As  it  cooled  off,  it  grew  smaller  and 
smaller,  and  its  velocity  of  rotation  increased  in  rapidity  by 
virtue  of  a  well- established  law  of  mechanics,  known  as 


498  ASTRONOMY. 

that  of  the  conservation  of  areas.  According  to  this  law, 
whenever  a  system  of  particles,  of  any  kind  whatever,  which 
is  rotating  around  an  axis,  changes  its  form  or  arrangement 
by  virtue  of  the  mutual  attractions  of  its  parts  among  them- 
selves, the  sum  of  all  the  areas  described  by  each  particle 
around  the  centre  of  rotation  in  any  unit  of  time  remains 
constant.  This  sum  is  called  the  areolar  velocity. 

If  the  diameter  of  the  mass  is  reduced  to  one  half,  sup- 
posing it  to  remain  spherical,  the  area  of  any  plane  passing 
through  its  centre  will  be  reduced  to  one  fourth,  because 
areas  are  in  proportion  to  the  square  of  the  diameters. 
In  order  that  the  areolar  velocity  may  then  be  the  same 
as  before,  the  mass  must  rotate  four  times  as  fast.  The 
rotating  mass  we  have  described  must  have  had  an  axis 
around  which  it  rotated,  and  therefore  an  equator  defined 
as  being  everywhere  90°  from  this  axis.  In  consequence 
of  the  increase  in  the  velocity  of  rotation,  the  centrifugal 
force  would  also  be  increased  as  the  mass  grew  smaller. 
This  force  varies  as  the  radius  of  the  circle  described  by 
the  particle  multiplied  by  the  square  of  the  angular  velocity. 
Hence  when  the  masses,  being  reduced  to  half  the  radius, 
rotate  four  times  as  fast,  the  centrifugal  force  at  the  equa- 
tor would  be  increased  ^  X  4a,  or  eight  times.  The  gravi- 
tation of  the  mass  at  the  surface,  being  inversely  as  the 
square  of  the  distance  from  the  centre,  or  of  the  radius, 
would  be  increased  four  times.  Therefore  as  the  masses 
continue  to  contract,  the  centrifugal  force  increases  at  a 
more  rapid  rate  than  the  central  attraction.  A  time  would 
therefore  come  when  they  would  balance  each  other  at  the 
equator  of  the  mass.  The  mass  would  then  cease  to  con- 
tract at  the  equator,  but  at  the  poles  there  would  be  no 
centrifugal  force,  and  the  gravitation  of  the  mass  would 
grow  stronger  and  stronger.  In  consequence  the  mass  would 
at  length  assume  the  form  of  a  lens  or  disk  very  thin  in  pro- 
portion to  its  extent.  The  denser  portions  of  this  lens 
would  gradually  be  drawn  toward  the  centre,  and  there 
more  or  less  solidified  by  the  process  of  cooling.  A  point 


COSMOGONY.  499 

would  at  length  be  reached,  when  solid  particles  would  begin 
to  be  formed  throughout  the  whole  disk.  These  would  grad- 
ually condense  around  each  other  and  form  a  single  planet,  or 
they  might  break  up  into  small  masses  and  form  a  group  of 
planets.  As  the  motion  of  rotation  would  not  be  altered 
by  these  processes  of  condensation,  these  planets  would  all 
be  rotating  around  the  central  part  of  the  mass,  which  is 
supposed  to  have  condensed  into  the  sun. 

It  is  supposed  that  at  first  these  planetary  masses,  being 
very  hot,  were  composed  of  a  central  mass  of  those  sub- 
stances which  condensed  at  a  very  high  temperature,  sur- 
rounded by  the  vapors  of  those  substances  which  were 
more  volatile.  We  know,  for  instance,  that  it  takes  a  much 
higher  temperature  to  reduce  lime  and  platinum  to  vapor 
than  it  does  to  reduce  iron,  zinc,  or  magnesium.  There- 
fore, in  the  original  planets,  the  limes  and  earths  would 
condense  first,  while  many  other  metals  would  still  be  in  a 
state  of  vapor.  The  planetary  masses  would  each  be 
affected  by  a  rotation  increasing  in  rapidity  as  they  grew 
smaller,  and  would  at  length  f  orm  masses  of  melted  metals 
and  vapors  in  the  same  way  as  the  larger  mass  out  of  which 
the  sun  and  planets  were  formed.  These  masses  would 
then  condense  into  a  planet,  with  satellites  revolving 
around  it,  just  as  the  original  mass  condensed  into  sun  and 
planets. 

At  first  the  planets  would  be  so  hot  as  to  be  in  a  molten 
condition,  each  of  them  probably  shining  like  the  sun. 
They  would,  however,  slowly  cool  off  by  the  radiation  of 
heat  from  their  surfaces.  So  long  as  they  remained  liquid, 
the  surface,  as  fast  as  it  grew  cool,  would  sink  into  the  in- 
terior on  account  of  its  greater  specific  gravity,  and  its 
place  would  be  taken  by  hotter  material  rising  from  the 
interior  to  the  surface,  there  to  cool  off  in  its  turn.  There 
would,  in  fact,  be  a  motion  something  like  that  which  occurs 
when  a  pot  of  cold  water  is  set  upon  the  fire  to  boil. 
Whenever  a  mass  of  water  at  the  bottom  of  the  pot  is 
heated,  it  rises  to  the  surface?  and  the  cool  water  moves 


500  ASTRONOMY. 

down  to  take  its  place.  Thus,  on  the  whole,  so  long  as 
the  planet  remained  liquid,  it  would  cool  off  equally 
throughout  its  whole  mass,  owing  to  the  constant  motion 
from  the  centre  to  the  circumference  and  back  again.  A 
time  would  at  length  arrive  when  many  of  the  earths  and 
metals  would  begin  to  solidify.  At  first  the  solid  particles 
would  be  carried  up  and  down  with  the  liquid.  A  time 
would  finally  arrive  when  they  would  become  so  large 
and  numerous,  and  the  liquid  part  of  the  general  mass 
become  so  viscid,  that  the  motion  would  be  obstructed. 
The  planet  would  then  begin  to  solidify.  Two  views 
have  been  entertained  respecting  the  process  of  solidifica- 
tion. 

According  to  one  view,  the  whole  surface  of  the  planet 
would  solidify  into  a  continuous  crust,  as  ice  forms  over  a 
pond  in  cold  weather,  while  the  interior  was  still  in  a 
molten  state.  The  interior  liquid  could  then  no  longer 
come  to  the  surface  to  cool  off,  and  could  lose  no  heat 
except  what  was  conducted  through  this  crust.  Hence 
the  subsequent  cooling  would  be  much  slower,  and  the 
globe  would  long  remain  a  mass  of  lava,  covered  over  by 
a  comparatively  thin  solid  crust  like  that  on  which  we 
live. 

The  other  view  is  that,  when  the  cooling  attained  a  cer- 
tain stage,  the  central  portion  of  the  globe  would  be 
solidified  by  the  enormous  pressure  of  the  superincumbent 
portions,  while  the  exterior  was  still  fluid,  and  that  thus 
the  solidification  would  take  place  from  the  centre  out- 
ward. 

It  is  still  an  unsettled  question  whether  the  earth  is  now 
solid  to  its  centre,  or  whether  it  is  a  great  globe  of  molten 
matter  with  a  comparatively  thin  crust.  Astronomers  and 
physicists  incline  to  the  former  view  ;  geologists  to  the 
latter  one.  Whichever  view  may  be  correct,  it  appears 
certain  that  there  are  great  lakes  of  lava  in  the  interior 
from  which  volcanoes  are  fed. 

It  must  be  understood  that  the  nebular  hypothesis,  as 


COSMOGONY.  501 

we  have  explained  it,  is  not  a  perfectly  established  scien- 
tific theory,  but  only  a  philosophical  conclusion  founded 
on  the  widest  study  of  nature,  and  pointed  to  by  many 
otherwise  disconnected  facts.  The  widest  generalization 
associated  with  it  is  that,  so  far  as  we  can  see,  the  universe 
is  not  self-sustaining,  but  is  a  kind  of  organism  which,  like 
all  other  organisms  we  know  of,  must  come  to  an  end  in 
consequence  of  those  very  laws  of  action  which  keep  it 
going.  It  must  have  had  a  beginning  within  a  certain 
number  of  years  which  we  cannot  yet  calculate  with  cer- 
tainty, but  which  cannot  much  exceed  20,000,000,  and  it 
must  end  in  a  chaos  of  cold,  dead  globes  at  a  calculable 
time  in  the  future,  when  the  sun  and  stars  shall  have 
radiated  away  all  their  heat,  unless  it  is  re-created  by  the 
action  of  forces  of  which  we  at  present  know  nothing. 


FINIS. 


INDEX. 


'HIS  index  is  intended  to  point  out  the  subjects  treated  in  the 
work,  and  further,  to  give  references  to  the  pages  where  technical  terms 
are  denned  or  explained. 


Aberration-constant,  values  of, 
244. 

Aberration  of  a  lens  (chromatic), 
60. 

Aberration  of  a  lens  (spherical), 
61. 

Aberration  of  light,  238. 

Absolute  parallax  of  stars  denned, 
476. 

Accelerating  force  defined,  140. 

Achromatic  telescope  described, 
60. 

ADAMS'S  work  on  perturbations  of 
Uranus,  366. 

Adjustments  of  a  transit  instru- 
ment are  three  ;  for  level,  for 
collimation,  and  for  azimuth,  77. 

Aerolites,  375. 

AIRY'S  determination  of  the  densi- 
ty of  the  earth,  193. 

Algol  (variable  star),  440. 

Altitude  of  a  star  defined,  25. 

Annular  eclipses  of  the  sun,  175. 

Autumnal  equinox,  110. 

Apparent  place  of  a  star,  235. 

Apparent  semi -diameter  of  a  celes- 
tial body  defined,  52. 

Apparent  time,  260. 

ARAGO'S  catalogue  of  Aerolites, 
375. 

Arc  converted  into  time,  32. 


ARGELANDER'S  Durchmusterung, 
435. 

ARGELANDER'S  uranometry,  435. 

ARISTARCHUS  determines  the  solar 
parallax,  223. 

ARTSTARCHUS  maintains  the  rota- 
tion of  the  earth,  14. 

Artificial  horizon  used  with  sex- 
tant on  shore,  95. 

Aspects  of  the  planets,  272. 

ASTEN'S,  VON,  computation  of 
orbit  of  Donati's  comet,  409. 

Asteroids  defined,  268. 

Asteroids,  number  of,  200  in  1879, 
341. 

Asteroids,  their  magnitudes,  341. 

Astronomical  instruments  (in  gen- 
eral), 53. 

Astronomical  units  of  length  and 
mass,  214. 

Astronomy  (defined),  1. 

Atmosphere  of  the  moon,  331. 

Atmospheres  of  the  planets,  see 
Mercury,  Venus,  etc. 

Axis  of  the  celestial  sphere  de- 
fined, 23. 

Axis  of  the  earth  defined,  25. 

Azimuth  error  of  a  transit  instru- 
ment, 77. 

BAILY'S  determination  of  the  den- 
sity of  the  earth,  192. 


504 


INDEX. 


BAYER'S  uranometry  (1654),  420. 

BEER  and  MAEDLER'S  map  of 
the  moon,  333. 

BESSEL'S  parallax  of  61  Cygnf 
(1837),  476. 

BESSEL'S  work  on  the  theory  of 
Uranus,  366. 

BIELA'S  comet,  404. 

Binary  stars,  450. 

Binary  stars,  their  orbits,  452. 

BODE'S  catalogue  of  stars,  435. 

BODE'S  law  stated,  269. 

BOND'S  discovery  of  the  dusky 
ring  of  Saturn,  1850,  356. 

BOND'S  observations  of  Donati's 
comet,  392. 

BOND'S  theory  of  the  constitution 
of  Saturn's  rings,  359. 

BOUVARD'S  tables  of  Uranus,  365. 

BRADLEY  discovers  aberration  in 
1729,  240. 

BRADLEY'S  method  of  eye  and  ear 
observations  (1750),  79. 

Brightness  of  all  the  stars  of  each 
magnitude,  439. 

Calendar,  can  it  be  improved  ? 
261. 

Calendar  of  the  French  Republic, 
262. 

Calendars,  how  formed,  248. 

CALLYPUS,  period  of,  290. 

Cassegrainian  (reflecting)  telescope, 
67. 

CASSINI  discovers  four  satellites  of 
Saturn  (1684-1671),  360. 

CASSINI'S  value  of  the  solar  paral- 
lax, 9 "'5,  226. 

Catalogues  of  stars,  general  ac- 
count, 434. 

Catalogues  of  stars,  their  arrange- 
ment, 265. 

CAVENDISH,  experiment  for  deter- 
mining the  density  of  the  earth, 
192. 

Celestial  mechanics  defined,  3. 

Celestial  sphere,  14,  41. 

Central  eclipse  of  the  sun,  177. 


Centre  of  gravity  of  the  solar  sys- 
tem, 272. 

Centrifugal  force,  a  misnomer, 
210. 

CHRISTIE'S  determination  of  mo- 
tion of  stars  in  line  of  sight,  471. 

Chromatic  aberration  of  a  lens,  60. 

Chronograph  used  in  transit  ob- 
servations, 19. 

Chronology,  245. 

Chronometers,  70. 

CLAIRAUT  predicts  the  return  of 
Halley's  comet  (1759),  397. 

CLARKE'S  elements  of  the  earth, 
202. 

Clocks,  70. 

Clusters  of  stars  are  often  formed 
by  central  powers,  464. 

Coal-sacks  in  the  milky  way,  415, 
485. 

Coma  of  a  comet,  388. 

Comets  denned,  268. 

Comets  formerly  inspired  terror, 
405-6. 

Comets,  general  account,  388. 

Comets'  orbits,  theory  of,  400. 

Comets'  tails,  388. 

Comets'  tails,  repulsive  force,  395. 

Comets,  their  origin,  401. 

Comets,their  physical  constitution, 
393. 

Comets,  their  spectra,  393. 

Conjunction  (of  a  planet  with  the 
sun)  denned,  114. 

Collimation  of  a  transit  instru- 
ment, 77. 

Conjugate  foci  of  a  lens  defined, 
65. 

Constellations,  414. 

Constellations,  in  particular,  422, 


Construction  of  the  Heavens,  478. 
Co-ordinates  of  a  star  defined,  41. 
COPELAND  observes  spectrum  of 

new  star  of  1876,  445. 
CORNU'S  observations  of  spectrum 

of  new  star  of  1876,  445. 


INDEX. 


505 


CORNU  determines  the  velocity  of 
light,  222. 

Correction  of  a  clock  denned,  72. 

Cosmical  physics  defined,  3. 

Cosmogony  defined,  492. 

Corona,  its  spectrum,  305. 

Corona  (the)  is  a  solar  appendage, 
302. 

Craters  of  the  moon,  328. 

Day,  how  subdivided  into  hours, 
etc.,  257. 

Days,  mean  solar,  and  solar,  259. 

Declination  of  a  star  defined,  20. 

Dispersive  power  of  glass  defined, 
81. 

Distance  of  the  fixed  stars,  412, 
474. 

Distribution  of  the  stars,  489. 

Diurnal  motion,  10. 

Diurnal  paths  of  stars  are  circles 
12. 

Dominical  letter,  255. 

DONATI'S  comet  (1858),  407. 

Double  (and  multiple)  stars,  44P. 

Double  stars,  their  colors,  452. 

Earth  (the),  a  sphere,  9. 

Earth  (the)  general  account  of,  188. 

Earth  (the)  is  a  point  in  compari- 
son with  the  distance  of  the  fixed 
stars,  17. 

Earth  (the)  is  isolated  in  space,  10. 

Earth's  annual  revolution,  98. 

Earth's  atmosphere  at  least  100 
miles  in  height,  380. 

Earth's  axis  remains  parallel  to  it- 
self during  an  annual  revolution, 
109,  110. 

Earth's  density,  188,  190. 

Earth's  dimensions,  201. 

Earth's  internal  heat,  493. 

Earth's  mass,  188. 

Earth's  mass  with  various  values 
of  solar  parallax  (table),  230. 

Earth's  motion  of  rotation  proba- 
bly not  uniform.  148. 

Earths'  (the)  relation  to  the  heav- 
ens, 9. 


Earth's  rotation  maintained  by 
ATIISTARCHUS  and  TIMOCHARIS, 
and  opposed  by  PTOLEMY,  14. 

Earth's  surface  is  gradually  cool- 
ing, 493. 

Eccentrics  devised  by  the  ancients 
to  account  for  the  irregularities 
of  planetary  motions,  121. 

Eclipses  of  the  moon,  170. 

Eclipses  of  the  sun  and  moon,  168. 

Eclipses  of  the  sun,  explanation, 
172. 

Eclipses  of  the  sun,  physical  phe- 
nomena, 297. 

Eclipses,  their  recurrence,  177. 

Ecliptic  defined,  100. 

Ecliptic  limits,  178. 

Elements  of  the  orbits  of  the  ma- 
jor planets,  276. 

Elliptic  motion  of  a  planet,  its 
mathematical  theory,  125. 

Elongation  (of  a  planet)  defined, 
114. 

ENCKE'S  comet,  409. 

ENCKE'S  value  of  the  solar  paral- 
lax, 8"  "857,  226. 

ENGELMANN'S  photometric  meas- 
ures of  Jupiter's  satellites,  350. 

Envelopes  of  a  comet,  390. 

Epicycles,  their  theory,  119. 

Equation  of  time,  258. 

Equator  (celestial)  defined,  19,  24. 

Equatorial  telescope,  description 
of,  87. 

Equinoctial  defined,  24. 

Equinoctial  year,  207. 

Equinoxes,  104. 

Equinoxes  ;  how  determined,  105. 

Evection,  moon's  163. 

Eye-pieces  of  telescopes,  62. 

Eye  (the  naked)  sees  about  2000 
stars,  411,  414. 

FABRITIUS  observes  solar  spots 
(1611),  288. 

Figure  of  the  earth,  198. 

FIZEAU  determines  the  velocity  of 
light,  222. 


506 


INDEX. 


FLAMSTEED'S  catalogue  of    stars 

(1689),  .431. 
Focal  distance  of  a  lens  defined, 

65. 
FOUCAULT  determines  the  velocity 

of  light,  222. 

Future  of  the  solar  system,  501. 
Galaxy,  or  milky  way,  415. 
GALILEO  observes  solarspots  (1611), 

288. 
GALILEO'S  discovery  of  satellites 

of  Jupiter  (1610),  343. 
GALILEO'S  resolution  of  the  milky 

way  (1610),  415. 
GALLE    first   observes     Neptune 

(1846),  367. 

Geodetic  surveys,  199. 
Golden  number,  252. 
GOULD'S  uranometry,  435. 
Gravitation  extends  to   the  stars, 

451,  456. 
Gravitation  resides  in  each  particle 

of  matter,  139. 
Gravitation,  terrestrial    (its  laws), 

194. 
Gravity    (on    the  earth)  changes 

with  the  latitude,  203. 
Greek  alphabet,  7. 
Gregorian  calendar,  255. 
GYLDEN,  hypothetical  parallax  of 

stars,  454. 
GYLDEN  on  the  distribution  of  the 

stars,  489. 
HALLEY  predicts  the  return  of  a 

comet  (1682),  397. 
HALLEY'S  comet,  398. 
HALL'S  discovery  of  satellites  of 

Mars,  338. 
HALL'S  rotation-period  of  Saturn, 

352. 
HARKNESS  observes  the  spectrum 

of  the  corona  (1869),  305. 
Hauptpunkte  of  an  objective,  64. 
HANSEN'S  value  of  the  solar  paral- 
lax, 8".92,  227. 
HEIS'S  uranometry  of  the  northern 

sky,  417. 


HELMHOLTZ'S   measures    of    the 

m    limits  of  naked  eye  vision,  4. 

HERSCHEL  (W.),  first  observes 
the  spectra  of  stars  (1798),  468. 

HERSCHEL  (W.),  discovers  two 
satellites  of  Saturn  (1789),  360. 

HERSCHEL  (W.),  discovers  two 
satellites  of  Uranus  (1787),  363. 

HERSCHEL  (W.)  discovers  Uranus 
(1781),  362. 

HERSCHEL  (W.)  observes  double 
stars  (1780),  452. 

HERSCHEL'S  catalogues  of  nebu- 
lae, 457. 

HERSCHEL'S  star-gauges,  479. 

HERSCHEL  (W.)  states  that  the 
solar  system  is  in  motion  (1783), 
474. 

HERSCHEL'S  (W.)  views  on  the 
nature  of  nebulae,  458. 

HEVELTUS'S  catalogue  of  stars,435. 

HILL'S  (G.  W.)  orbit  of  Donati's 
comet,  409. 

HILL'S  (G.  W.)  theory  of  Mer 
cury,  323. 

HOOKE'S  drawings  of  Mars  (1666), 
336. 

Horizon  (celestial — sensible)  of  an 
observer  defined,  23. 

HORROX'S  guess  at  the  solar  par- 
allax, 225. 

Hour  angle  of  a  star  defined,  25. 

HUBBARD'S  investigation  of  orbit 
of  Biela's  comet,  404. 

HOGGINS'  determination  of  mo- 
tion of  stars  in  line  of  sight,  471. 

HUGGINS  first  observes  the  spectra 
of  nebulae  (1864),  465. 

HUGGINS'  observations  of  the  spec- 
tra of  the  planets,  370,  et  seq. 

HUGGINS'  and  MILLER'S  observa- 
tions of  spectrum  of  new  star  of 
1866,  445-6. 

HUGGINS'  and  MILLER'S  observa 
tions  of  stellar  spectra,  468. 

HUYGHENS  discovers  a  satellite  of 
Saturn  (1655),  360. 


INDEX. 


507 


HUYGHENS  discovers  laws  of  cen- 
tral forces,  135. 

HUYGHENS    discovers    the     neb- 
ula of  Orion  (1650),  457. 
HUYGHENS'    explanation   of   the 
appearances  of    Saturn's  rings 
(1655),  356. 

HUYGHENS'  guess  at  the  solar  par- 
allax, 226. 

HUYGHENS'  resolution  of  the  milky 
way,  416. 

Inferior  planets  defined,  116. 

Intramercurial  planets,  322. 

JANSSEN  first  observes  solar  promi- 
nences in  daylight,  304. 

JANSSEN'S  photographs  of  the  sun, 
281. 

Julian  year,  250. 

Jupiter,  general  account,  343. 

Jupiter's  rotation  time,  346 

Jupiter !s  satellites,  346. 

Jupiter's  satellites,  their  elements, 
351. 

KANT'S  nebular  hypothesis,  492. 

KEPLER'S  idea  of  the  milky  way, 
416. 

Kepler's  laws  enunciated,  125. 

KEPLER'S  laws  of  planetary  mo- 
tion, 122. 

KLEIN,  photometric  measures  of 
Beta  LyrcR,  442. 

LACAILLE'S  catalogues  of  nebulae, 
457. 

LANGLEY'S  measures  of  solar  heat, 
283. 

LANGLEY'S  measures  of  the  heat 
from  sun  spots,  286. 

LAPLACE  investigates  the  accelera- 
tion of  the  moon's  motion, 
146. 

LAPLACE'S  nebular  hypothesis,  492. 

LAPLACE'S  investigation  of  the 
constitution  of  Saturn's  rings, 
359. 

LAPLACE'S  relations  between  the 
mean  motions  of  Jupiter's  satel- 
lites, 349. 


LASSELL  discovers  Neptune's  sat- 
ellite (1847),  369. 

LASSELL  discovers  two  satellites  of 
Uranus  (1847),  363. 

Latitude  (geocentric — geographic) 
of  a  place  on  the  earth  defined, 
203. 

Latitude  of  a  point  on  the  earth  is 
measured  by  the  elevation  of  the 
pole,  21. 

Latitudes  and  longitudes  (celes- 
tial) defined,  112. 

Latitudes  (terrestrial),  how  deter- 
mined, 47,  48. 

LA  SAGE'S  theory  of  the  cause  of 
gravitation,  150. 

Level  of  a  transit  instrument,  77. 

LE  VERRIER  computes  the  orbit  of 
meteoric  shower,  384. 

LE  VERRIER 's  researches  on  the 
theory  of  Mercury,  323. 

LE  VERRIER 's  work  on  perturba- 
tions of  Uranus,  366. 

Light-gathering  power  of  an  ob- 
ject glass,  56. 

Light- ratio  (of  stars)  is  about  2 '5, 
417. 

Line  of  collimation  of  a  telescope, 
59. 

Local  time,  32. 

LOCKYER'S  discovery  of  a  spec- 
troscopic  method,  304. 

Longitude  of  a  place  may  be  ex- 
pressed in  time,  33. 

Longitude  of  a  place  on  the  earth 
(how  determined),  34,  37,  38,  41. 

Longitudes  (celestial)  defined,  112. 

Lucid  stars  defined,  415. 

Lunar  phases,  nodes,  etc.  See 
Moon's  phases,  nodes,  etc. 

MAEDLER'S  theory  of  a  central 
sun,  478. 

Magnify  ing  power  of  an  eye -piece, 
55. 

Magnifying  powers  (of  telescopes), 
which  can  be  advantageously 
employed,  58. 


508 


INDEX. 


Magnitudes  of  the  stars,  416. 

Major  planets  denned,  268. 

Mars,  its  surface,  336. 

Mars,  physical  description,  334. 

Mars,  rotation,  336. 

Mars's    satellites     discovered    by 

HALL  (1877),  338. 
MARIUS'S  claim   to   discovery  of 

Jupiter's  satellites,  343. 
MASKELYNE  determines  the  den- 
sity of  the  earth,  192. 
Mass  and  density  of  the  sun  and 

planets,  277. 
Mass  of   the    sun  in   relation   to 

masses  of  planets,  227. 
Masses  of  the  planets,  232. 
MAXWELL'S  theory  of  constitution 

of  Saturn's  rings,  360. 
MAYER  (C.)  first  observes  double 

stars  (1778),  452. 
Mean  solar  time  defined,  28. 
Measurement  of  a  degree  on  the 

earth's  surface,  201. 
Mercury's  atmosphere,  314. 
Mercury,  its    apparent   motions, 

310. 
Mercury,  its  aspects  and  rotation, 

318. 

Meridian  (celestial)  defined,  21,  25. 
Meridian  circle,  83. 
Meridian  line  defined,  25. 
Meridians  (terrestrial)  defined,  21. 
MESSIER'S  catalogues  of  nebulae, 

457. 

Metonic  cycle,  251. 
Meteoric  showers,  380. 
Meteoric  showers,  orbits,  383. 
Meteors  and  comets,  their  relation, 

383. 
Meteors    first    visible   about   100 

miles  above  the  surface  of  the 

earth,  380. 

Meteors,  general  account,  375. 
Meteors,  their  cause,  377. 
Metric  equivalents,  8. 
MICHAELSON  determines   the  ve- 
locity of  light  (1879),  222. 


MICHELL'S  researches   on   distri- 
bution of  stars  (1777),  449. 
•Micrometer  (filar),  description  and 

use,  89. 

Milky  way,  415. 

Milky  way,  its  general  shape  ac- 
cording to  HERSCHEL,  480. 
Minimum    Visibile    of    telescope* 

(table),  419. 

Minor  planets  defined,  268. 
Minor  planets,    general  account, 

340. 

Mira  Ceti  (variable  star),  440. 
Mohammedan  calendar,  252. 
Months,  different  kinds,  249. 
Moon's  atmosphere,  331. 
Moon  craters,  329. 
Moon,  general  account,  326. 
Moon's  light  and  heat,  331. 
Moon's  light  l-618,000th  of   the 

sun's,  332. 
Moon's   motions    and   attraction, 

152. 

Moon's  nodes,  motion  of,  159.  • 
Moon's  perigee,  motion  of,  103. 
Moon's  phases,  154. 
Moon's  rotation,  164. 
Moon's  secular  acceleration,  146. 
Moon's  surface,  does  it   change, 

332. 

Moon's  surface,  its  character,  328. 
Motion    of    stars    in    the  line  of 

sight,  470. 
Mountains    on    the    moon    often 

7000  metres  high,  330. 
Nadir  of  an  observer  defined,  23. 
Nautical  almanac  described,  263. 
Nebulae  and  clusters,  how  distrib- 
uted, 465. 
Nebulaa    and  clusters  in  general, 

457. 
Nebula  of  Orion,  the  first  telescopic 

nebula  discovered  (1650),  457. 
Nebulae,  their  spectra,  465. 
Nebular  hypothesis  stated,  497. 
Neptune,  discovery  of  by  LE  VER- 

RIER  and  ADAMS  (1846),  367. 


INDEX. 


509 


Neptune,    general    account,  365. 

Neptune's  satellite,  elements,  309. 

New  star  of  1876  has  apparently 
become  a  planetary  nebula,  445. 

New  stars,  443. 

NEWTON  (I.)  calculates  orbit  of 
comet  of  1680,  406. 

NEWTON  (I.)  Laws  of  Force, 
134. 

Newtonian  (reflecting)  telescope, 
66. 

NEWTON'S  (I.)  investigation  of 
comet  orbits,  396. 

NEWTON'S  (H.  A.)  researches  on 
meteors,  386. 

NEWTON'S  (H.  A.)  theory  of  con- 
stitution of  comets,  394. 

Nucleus  of  a  comet,  388. 

Nucleus  of  a  solar  spot,  287. 

Nutation,  211. 

Objectives  (mathematical  theory), 
63. 

Objectives  or  object  glasses,  54. 

Obliquity  of  the  ecliptic,  106. 

Occultations  of  stars  by  the  moon 
(or  planets),  186. 

OLBERS'S  hypothesis  of  the  origin 
of  asteroids,  340,  342. 

OLBERS  predicts  the  return  of  a 
meteoric  shower,  381. 

Old  style  (in  dates),  254. 

Opposition  (of  a  planet  to  the  sun) 
defined,  115. 

Oppositions  of  Mars,  335. 

Parallax  of  Mars,  220,  221. 

Parallax  of  the  sun,  216. 

Penumbra  of  the  earth's  or  moon's 
shadow,  174. 

Photosphere  of  the  sun,  279. 

PICARD  publishes  the  Connaissance 
des  Terns  (1679),  263. 

PICKERING'S  measures  of  solar 
light,  283. 

Planets,  their  relative  size  exhib- 
ited, 269. 

POUILLET'S  measures  of  solar  radi- 
ation, 285. 


Precession  of  the  equinoxes,  206, 
209. 

PTOLEMY  determines  the  solar 
parallax,  225. 

Parallax  (annual)  defined,  50. 

Parallax  (equatorial  horizontal)  de- 
fined, 52. 

Parallax  (horizontal)  defined,  50. 

Parallax  (in  general)  defined,  50. 

Parallel  sphere  defined,  26. 

Parallels  of  declination  defined,  24. 

Parallax  of  the  stars,  general  ac- 
count, 476. 

PEIRCE'S  theory  of  the  constitu- 
tion of  Saturn's  rings,  359. 

Pendulums  of  astronomical  clocks, 
71. 

Periodic  comets,  elements,  399. 

Perturbations  defined,  144. 

Perturbations  of  comets  by  Jupi- 
ter, 403. 

Photometer  defined,  417. 

PIAZZI  discovers  the  first  asteroid 
(1801),  340. 

Planetary  nebulaB  defined,  459. 

Planets  ;  seven  bodies  so  called  by 
the  ancients,  96. 

Planets,  their  apparent  and  real 
motions,  113. 

Planets,  their  physical  constitu- 
tion, 370. 

Pleiades,  map  of,  425. 

Pleiades,  these  stars  are  physically 
connected,  449. 

Polar  distance  of  a  star,  25. 

Poles  of  the  celestial  sphere  de- 
fined, 14,  20,  24. 

Position  angle  defined,  90,  450. 

Power  of  telescopes,  its  limit,  328. 

Practical  astronomy  (defined),  2. 

Prime  vertical  of  an  observer  de- 
fined, 25. 

Problem  of  three  bodies,  141. 

PROCTOR'S  map  of  distribution  of 
nebulae  and  clusters,  466. 

PROCTOR'S  rotation  period  of  Mars, 
336. 


510 


INDEX. 


Proper  motions  of  stars,  472. 

Proper  motion  of  the  sun,  473. 

PTOLEMY'S  catalogue  of  stars, 
435. 

PTOLEMY  maintains  the  immova- 
bility of  the  earth,  14. 

PYTHAGORAS'  conception  of  crys- 
talline spheres  for  the  planets,  96. 

Radiant  point  of  meteors,  381. 

Rate  of  a  clock  defined,  72. 

Reading  microscope,  81,  85. 

Red  stars  (variable  stars  often  red), 
442. 

Reflecting  telescopes,  66. 

Reflecting  telescopes,  their  advan- 
tages and  disadvantages,  68,  69. 

Refracting  telescopes,  53. 

Refraction  of  light  in  the  atmos- 
phere, 234. 

Refractive  power  of  a  lens  defined, 
65. 

Refractive  power  of  glass  defined, 
61. 

Relative  parallax  of  stars  defined, 
476. 

Resisting  medium  in  space,  409. 

Reticle  of  a  transit  instrument,  76. 

Retrogradations  of  the  planets  ex- 
plained, 118. 

Right  ascension  of  a  star  defined, 
22. 

Right  ascensions  of  stars,  how 
determined  by  observation,  31. 

Right  sphere  defined,  27. 

Eilkn  on  the  moon,  330. 

ROEMER  discovers  that  light  moves 
progressively,  239. 

ROSSE'S  measure  of  the  moon's 
heat,  332. 

Saros  (the),  181. 

Saturn,  general  account,  352. 

Saturn's  rings,  354. 

Saturn's  rings,  their  constitution, 
359. 

Saturn's  rings,  their  phases,  357. 

Saturn's  satellites,  360. 

Saturn's  satellites,  elements,  361. 


SAVARY  first  computes  orbit  of  a 
binary  star  (1826),  456. 

.SCHIAPARELLI'S  theory  of  rela- 
tions of  comets  and  meteors, 
385. 

SCHMIDT  discovers  new  star  in 
Cygnus  (1876),  445. 

SCHMIDT'S  observations  of  new 
star  of  1866,  444. 

SCHOENPELD'S  Durchmusterung, 
436. 

SCHROETER'S  observations  on  the 
rotation  of  Venus,  316. 

SCHW ABE'S  observations  of  sun 
spots,  293. 

Seasons  (the),  108. 

SECCHI'S  estimate  of  solar  tempera- 
ture 6,100,000°  C.,  286. 

SECCHI'S  types  of  star  spectra, 
468. 

Secondary  spectrum  of  object 
glasses  defined,  62. 

Seconds  pendulum,  length,  formu- 
la for  it,  204. 

Secular  acceleration  of  the  moon's 
mean  motion,  146. 

Secular  perturbations  defined,  145. 

Semi- diameters  (apparent)  of  ce- 
lestial objects  defined,  52. 

Semi-diurnal  arcs  of  stars,  45. 

Sextant,  92. 

Shooting  stars,  377. 

Sidereal  system,  its  shape  accord- 
ing to  HERSCHEL,  484. 

Sidereal  time  explained,  29. 

Sidereal  year,  207. 

Signs  of  the  Zodiac,  105. 

Silvered  glass  reflecting  telescopes, 
69. 

Sirius  is  about  500  times  brighter 
than  a  star  6m,  418. 

Stars  had  special  names  3000 
B.C.,  420. 

Solar  corona,  extent  of,  299. 

Solar  cycle,  255. 

Solar  heat  and  light,  its  cause,  306. 

Solar  heat,  its  amount,  284. 


INDEX. 


511 


Solar  motion  in  space,  473. 

Solar  parallax  from  lunar-inequali- 
ty, 223. 

Solar  parallax  from  Mars,  220. 

Solar  parallax  from  velocity  of 
light,  222. 

Solar  parallax,  history  of  attempts 
to  determine  it,  223. 

Solar  parallax,  its  measures,  216. 

Solar  parallax  probably  about 
8" '81,  223. 

Solar  prominences  are  gaseous, 
303. 

Solar  system  defined,  97. 

Solar  system,  description,  267. 

Solar  system,  its  future,  309,  501. 

Solar  temperature,  286. 

Solstices,  103,  104. 

Spherical  aberration  of  a  lens,  61. 

Spherical  astronomy  (defined),  2. 

Spiral  nebulae  defined,  459. 

Star  clusters,  462. 

Star-gauges  of  HERSCIIEL,  479. 

Star  magnitudes,  416. 

Stars  of  various  magnitudes,  how 
distributed,  436-7. 

Stars  seen  by  the  naked  eye,  about 
2000,  411-414. 

Stars,  their  proper  motions,  472. 

Stars,  their  spectra,  468. 

STRUVE'S  (W.)  idea  of  the  distri- 
bution of  the  stars,  487. 

STHUVE'S  (W.)  parallax  of  alpha 
Lyra  (1838),  476. 

STRUVE'S  (W.)  search  for  [Nep- 
tune], 366. 

STRUVE'S  (O.)  supposition  of 
changes  in  Saturn's  rings,  358. 

SUFI'S  uranometry,  443. 

Summer  solstice,  110. 

Sun's  apparent  path,  101. 

Sun's  attraction  on  the  moon 
(and  earth),  156. 

Sun's  constitution,  305. 

Sun's  density,  230. 

Sun's  (the)  existence  cannot  be  in- 
definitely long,  495. 


Sun's  mass  over  700  times  that  of 

the  planets,  272. 
Sun's    motion  among   the    stars, 

101. 

Sun,  physical  description,  278. 
Sun's  proper  motion,  473. 
Sun's  rotation  time,  about  25  days, 

290. 

Sun-spots  and  faculae,  287. 
Sun-spots  are  confined  to  certain 

parts  of  the  disc,  289. 
Sun-spots,  cause  of  their  periodic 

appearance  unknown,  294. 
Sun's  surface  is  gradually  cooling, 

494. 

Sun-spots,  their  nature,  290. 
Sun-spots,  their  periodicity,  292. 
Superior  planets  (defined),  116. 
SWEDENBORG'S  nebular  hypothe- 
sis, 492. 
SWIFT'S    supposed    discovery    of 

Vulcan,  323. 

Symbols  used  in  astronomy,  6,  7. 
Telescopes,  their  advantages,  57, 

58. 

Telescopes  (reflecting),  66. 
Telescopes  (refracting),  53. 
TEMPEL'S  comet,  its  relation  to 

November  meteors,  384. 
Temporary  stars,  443. 
Theoretical  astronomy  (defined),  3. 
Tides,  165. 

Time  converted  into  arc,  32. 
TIMOCHARIS    maintains  the  rota- 
tion of  the  earth,  14. 
Total  solar  eclipses,  description  of, 

297. 

Transit  instrument,  74. 
Transit    instrument,   methods    of 

observation,  78. 
Transits  of  Mercury  and  Venus, 

318. 

Transits  of  Venus,  216. 
Triangulation,  199. 
Tropical  year,  207. 
Tvcno  BRAHE'S  catalogue  of  stars, 

435. 


512 


INDEX. 


TYCHO  BRAHE  observes  new  star 
of  1572,  443. 

Units  of  mass  and  length  employed 

in  astronomy,  213. 
Universal   gravitation    discovered 
by  NEWTON,  149. 

Universal      gravitation      treated, 
131. 

Universe    (the)   general    account, 
411. 

Uranus,  general  account,  362. 

Variable  and  temporary  stars,  gen- 
eral account,  440. 

Variable  stars,  440. 

Variable  stars,  their  periods,  442. 

Variable  stars,  theories  of,  445. 

Variation,  moon's,  163. 

Velocity  of  light,  244. 

Venus's  atmosphere,  317. 

Venus,  its  apparent  motions,  310. 

Venus,    its   aspect    and    rotation, 
315. 

Vernal  equinox,  102,  110. 

Vernier,  82. 

VOGEL'S  determination  of  motion 
of  stars  in  line  of  sight,  471. 

VOGEL'S  measures  of  solar  actinic 
force,  283. 

VOGEL'S    observations    of    Mer- 
cury's spectrum,  314. 


VOGEL'S  observations  of  spectrum 
of  new  star  of  1876,  445. 

VOGEL'S  observations  of  the  spec- 
tra of  the  planets,  370,  et  seq. 

Volcanoes  on  the  moon  supposed 
to  exist  by  HERSCHEL,  332. 

Vulcan,  322. 

WATSON'S  supposed  discovery  of 
Vulcan,  323,  324. 

Wave  and  armature  time,  40. 

Weight  of  a  bod}r  denned,  189. 

WILSON'S  theory  of  sun-spots,  290. 

Winter  solstice,  109. 

WOLF'S  researches  on  sun-spots, 
295. 

Years,  different  kinds,  250. 

YOUNG  observes  the  spectrum  of 
the  corona  (1869),  305. 

Zenith  denned,  19,  23. 

Zenith  telescope  described,  90. 

Zenith  telescope,  method  of  observ- 
ing, 92, 

Zodiac,  105. 

ZOELLNER'S  estimate  of  relative 
brightness  of  sun  and  planets, 
271. 

ZOELLNER'S  measure  of  the  rela- 
tive brightness  of  sun  and  moon, 
332. 

Zone  observations,  85. 


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